CiniSIlT SUES
quanta
of eancepts
R W. Atkins
Oxford Chemistry Series
General Editors
P. W. ATKINS J. S. E. HOLKER A. K. HOLLIDAY
Oxford Chemistry Series
1972
1. K. A. McLauchlan: Magnetic resonance
2. J. Robbins: Ions in solution (2): an introduction to electrochemistry
3. R. J. Puddephatt: The periodic table of the elements
4. R. A. Jackson: Mechanism: an introduction to the study of organic reactions
1973
5. D. Whittaker: Stereochemistry and mechanism
6. G. Hughes: Radiation chemistry
7. G. Pass : Ions in solution (3) : inorganic properties
8. E. B. Smith: Basic chemical thermodynamics
9. C. A. Coulson : The shape and structure of molecules
10. J. Wormald: Diffraction methods
11. J. Shorter: Correlation analysis in organic chemistry: an introduction to linear freeenergy relationships
12. E. S. Stern (ed): The chemist in industry (I): fine chemicals for polymers
13. A. Earnshaw and T. J. Harrington: The chemistry of the transition elements
1974
14. W. J. Albery : Electrode kinetics
16. W. S. Fyfe: Geochemistry
17. E. S. Stern (ed): The chemist in industry (2): human health and plant protection
18. G. C. Bond: Heterogeneous catalysis: principles and applications
19. R. P. H. Gasser and W. G. Richards: Entropy and energy levels
20. D. J. Spedding: Air pollution
21. P. W. Atkins: Quanta: a handbook of concepts
22. M. J. Pilling: Reaction kinetics
P. W. ATKINS
FELLOW OF LINCOLN COLLEGE. OXFORD
Quanta
a handbook of concepts
Clarendon Press ■ Oxford ■ 1974
Oxford University Press, Ely House, London W. 1
GLASGOW NEW YORK TORONTO MELBOURNE WELLINGTON
CAPE TOWN IBADAN NAIROBI DAR ES SALAAM LUSAKA ADDIS ABABA
DELHI BOMBAY CALCUTTA MADRAS KARACHI LAHORE DACCA
KUALA LUMPUR SINGAPORE HONG KONG TOKYO
ASEBOUND ISBN 19 855493 1
PAPERBACK ISBN 19 855494 X
TEXT © OXFORD UNIVERSITY PRESS 1974
ILLUSTRATIONS © P. W. ATKINS 1974
All rights reserved. No part of this publication may be reproduced,
stored in a retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording or otherwise, without
the prior permission of Oxford University Press
ACCESSION no.
l^
PRINTED IN GREAT BRITAIN
BY FLETCHER & SON LTD., NORWICH
Preface
Here is a book that attempts to explain the quantum theory
without mathematics.
Of course, I agree that quantum theory has an inescapable
mathematical structure; I agree that the full precision of the
theory, and its richness, can be conveyed only in the language
of mathematics; I also agree that to make substantial con
tributions to quantum theory one needs a firm grasp of its
mathematical basis. Nevertheless, I also believe that math
ematical precision is not what everyone is after— not everyone
has that kind of interest, nor indeed that amount of time.
Most of us have our pictures of the concepts of quantum
theorywe have some way of visualizing orbitals, transitions,
etc.and those of us who do research are more often than not
guided in broad outline by some visualizable model of the
system we are trying to describe. This book contains my ways
of thinking about the concepts of quantum theory. The
pictures, like all analogies and models, are only a partial
representation of the true situation; but I hope they contain
the heart of the matter, and enable the reader to understand
each idea at a physical, rather than mathematical, level.
I hope that a broad selection of people will find the book
useful. I have had in mind both the student of chemistry who
at all levels of his studies encounters unfamiliar, little
understood, or halfforgotten concepts, and his teacher who
is pressed for an explanation. I hope that by reference to this
book all will be provided with just enough information to
make the concept clear and perhaps even to be stimulated to
find our more. If my explanation is inadequate, my bibli
ography will direct the dissatisfied along a trail of others'
explanations.
Each entry is intended to explain, in plain language, the
physical content of its topic. Most are illustrated by examples,
and, where appropriate, counter examples. Where the entry
draws on information contained elsewhere in the book, or
where further development of a topic is desirable, or where the
reader might be felt to require more background, I have
labelled a word with an unobtrusive * to signify that it is in
the book. For a good reason I cannot now remember, the
label precedes the indicated word; the labelled word does not
always correspond to an exact entry, but its sense should be
sufficient to indicate the appropriate entry without ambi
guity. Unlike most dictionaries this has an index: some
concepts are buried inside others.
Where I feel it desirable that some mathematics be intro
duced (for example when it seems helpful to have a collection
of formulae to hand, e.g. in "perturbation theory), I have used
a system of Boxes and Tables. A Box is part of the text, and
contains handy formulae: these are confined to Boxes in order
that they be present but held away from the description in the
text. A Table contains more detailed information (sometimes
of mathematical expressions not required for a reading of the
text) which I judge it helpful to have in a volume of this kind
(so that it can be used at more than just its principal level of
qualitative description), and which I judge would have intruded
too vigorously into the main text. These Tables, which are
collected at the end of the volume, also contain a selection of
experimental data. I think it important that one has an idea of
the size of physical quantities, and their trends, and so these
numbers are included to sketch out the range of experimental
data. The Tables are by no means exhaustive: they simply peg
out the terrain.
Nearly every entry is followed by Questions. These are set
with a double purpose and are graded from the trivial to the
slightly tough. The early parts of each are intended to focus
the reader's mind on key points made in the preceding entry.
Most of these simple questions can be answered by referring to
the entry. The second purpose is to bring a tiny amount of
mathematics into the book in an unobtrusive way: the harder
parts of the questions (which when they occur are often
labelled 2) invite the reader to make his own mathematical
exposition of the entry, and contain hints and guidance to that
end. The answers to the questions are mostly in the text, in a
Box, or on a Table; if that is not so. Further information points
the way.
Each entry contains a section headed Further information.
This is a guide to the literature, and in it will be found refer
ences to books and articles where the reader can turn for more
information, the development of the topic, and the absent
mathematics. This section is not exhaustive (it contains works
I have found helpful) but once a trail is indicated the literature
is more easily penetrated. In this section I have attempted to
list the books in order of complexity so that the reader can
make a progression through difficulty. There are however,
two important exceptions to this organization. The first is the
reference to MQM. This is my Molecular quantum mechanics
(Clarendon Press, 1970); many of the topics are treated in
more detail in that book, and a reference to the appropriate
section is included at the head of each Further information.
The second exception reflects the fact that this book is a part
of the Oxford Chemistry Ser/'esalthough its size makes it a
sport— and I have made a point of directing the reader to the
other books in the series where the topics are developed: these
are denoted OCSn and listed together at the start of the bibli
ography. OPSn books are the first few of the analogous
Oxford Physics Series, and are listed likewise.
In a few places I have aimed at a higher level of exposition
than the rest, especially when I have been unable to give what
I consider to be a satisfactory explanation at the low, quali
tative level at which in general I have aimed. Let me stress,
however, that I do not use the word 'low' pejoratively: it is, I
believe, as important for a chemist to have a physical intuition
about the behaviour of submicroscopic phenomena as it is for
him to be able to manipulate the mathematics of the descrip
tion. In my 'low' level descriptions I am attempting to train
this intuition. I shall not draw attention to the entries I con
sider to be at this higher level: if they are not noticed, so much
the better. If they are noticed, and found bewildering, the ° on
keywords will guide the reader to quieter waters. In this con
nexion, however, I must draw attention to one distressing
feature of Nature. As in a conventional dictionary, where the
unlikely concepts of aardvark, aasvogel, and ablet bring con
fusion and difficulty in unnatural proportion into A, so by the
same quirk does Nature concentrate difficulty into the A of
Quanta. I find the entries of A more difficult than the entries
of B; perhaps you will too. (My message is that reading A— as a
browser might feel inclinedgives, I think, a false impression
of the overall level of the entries.)
Two other points are worth making. The first is an excuse.
In an attempt to keep the price of the book low (at least by
the standards we are being trained to expect) all the diagrams
are my own sketches. All were done on a balcony in Italy one
summer, and each jolt, smudge, or splash tells its story of
inquisitive mosquito, local ant, or homing wasp. Please forgive
their generally amateur appearance. The second point concerns
those to whom notes (such as these entries) are anathema,
those who want material for more books, or those who
seek an essay. Each should notice that the approximately
200 entries can be permuted in some 10 300 different ways:
surely out of these some decent books can be wrought?
The real purpose of writing a preface is to come to the part
that gives most pleasure: the thanking of all those who have
contributed to the production of the book and easing its grim
labour. A quite outstanding contributionbeyond the call of
duty and reasonable expectationhas been made by the
Clarendon Press and its anonymous officers. Their assistance
ranged from advice and help from the drawing office,
through detailed and lengthy discussion of presentation,
to careful and ingenious production. Two of my research
students deserve my thanks: Michael Clugston bent his per
ceptive eye on the proofs and saved me from much shame, and
John Roberts spent time coaxing contours out of computers;
to both am I most grateful. To the others I express my thanks
for accepting neglect without overt complaint. The typing of
the whole obscure manuscript (as my own typed original is
better regarded) was done briskly and efficiently by
Mrs. E. Price and Mrs. M. Long, both of whom deserve at
least what immortality this page can provide.
P.W.A.
A
ab initio. Not the whole of quantum chemistry is conducted
in Latin; the small portion to which it has by convention been
confined, as an ironic meeting of cultures, belongs to those
whose business is computers. Ab initio, roughly translated,
means from scratch, and is applied to the molecularstructure
computations that abhor the inclusion of empirical data and
attempt to calculate from first principles, which for our pur
pose are the "Schrodinger equation and the method of •self
consistent fields.
Further information. Richards and Horsley (1970} have
prepared a short, simple guide to ab initio calculations, and
work through a number of examples. They also discuss the
relation of such calculations to the semiempirical methods in
which approximations and empirical data are introduceo into
"selfconsistent field calculations.
adiabatic process. The term adiabatic is used in both thermo
dynamics and quantum mechanics, and the uses are analogous.
In the former it signifies that the process is occurring without
exchange of heat with the environment {as this implies that
there is no change of entropy, the process is often called
isentropic). In the latter it signifies that a change is occurring
so that the system makes no transition to other states.
Consider, for example, a hypothetical 'hydrogen atom
with a variable nuclear charge. If the atom is initially in its
ground state, and the nuclear charge is increased extremely
slowly, the electron will be sucked in closer to the nucleus;
but the atom remains in its ground state, and by the time
that Z= 2 the system is a ground state He + ion: this is there
fore an adiabatic process. Conversely, if the nuclear charge of
an atom is changed suddenly (for example, by the emission of
an electron in /3decay), the bound electron finds itself in a
different nuclear potential but with its original spatial distri
bution: this distribution can be expressed as a mixture of
°wavef unctions of the new atom, and so in this impulsive or
nonadiabatic transition the system is knocked into a range of
states of the final system. The slow compression of a "particle
confined in a square well is another example of an adiabatic
transition, for if the system is in the nth level of the original
box it will be in the nth level of the new, smaller box if the
compression is infinitely slow.
Further information. See MQM Chapter 7. An account of
adiabatic transitions in terms of "perturbation theory will be
found in §76 of Davydov (1965), who derives the condition
that a motion is adiabatic if the perturbation V(t) changes so
slowly that d V/dt is much smaller that the energy separations
in the vicinity of the initially occupied state: jdV/dr <§
(A£ } 2 /h. See also §50 of Bohm (1951 ) for a pleasing discussion
with straightforward mathematics. Recent review articles on
adiabatic and nonadiabatic processes in molecules have been
published by Ko/os (1970) and Nikitin (1970) respectively.
See "noncrossing rule. Thermodynamic adiabaticity is dis
cussed by Smith in his Basic chemical thermodynamics
(OCS 8) and entropy is related to the distribution of particles
among energy levels in Gasser and Richards' Entropy and
energy levels (OCS 19). This is the key connexion between
the thermodynamic and quantummechanical uses of the term
2 alternant hydrocarbon
adiabatic: an unchanged distribution of particles among states
is an isentropic situation.
alternant hydrocarbon. A chain of carbon atoms in a
hydrocarbon that can be labelled alternately by a Star, no star,
a star, and so on, such that no two stars are neighbours nor
two unstarred atoms are neighbours when the labelling is
complete, is an alternant hydrocarbon. An example is propene,
which could be labelled C*~CC*, or CC*~C; another
example is benzene (1), and an example of a nonalternant
hydrocarbon is azulene
(2). Alternants are distinguished by
several electronic characteristics:
(a) To every bonding orbital of energy — E there is a comp
lementary 'antibonding orbital of energy +E: the bonding
and antibonding orbitals are arranged symmetrically about
zero energy (Fig. A1).
(b)ln an evenalternant the number N of carbon atoms is
even, and usually the numbers* of starred atoms is equal to
FIG. Al. Energy levels ol 3 typical alternant (benzene) and a typical
nonalternant (cyclopentadienyl). Note the symmetric disposition of the
levels in the former.
the number n of unstarred. In this case there are ^N bonding
orbitals and ^/v antibonding orbitals. If n* and n are unequal
there are In*— nl nonbonding orbitals inserted between a
symmetrical array of n bonding and n antibonding orbitals
(forn <n*). In an oddalternant the number of carbon atoms
is odd, and usually there is one more starred atom than un
starred (n* — d+1 1, and one nonbonding orbital is inserted
between the symmetrical array of n bonding and n antibonding
orbitals {n+n * ^/V).
(c)The distribution of electrons is more uniform in altern
ant hydrocarbons than in nonalternants. This property is
expressed quantitatively by the CoutsonRushbrooke theorem
which states that the 7relectron "charge density on every atom
in the ground state of an alternant hydrocarbon is unity {each
carbon has just one TTelectron associated with it).
(d) In oddalternants the electron density of the correspond
ing cation or anion may be deduced very simply by considering
the form of the nonbonding orbital, for it is this orbital from
which an electron is taken to form the cation, or to which one
is added to give the anion, and in the neutral hydrocarbon the
charge, distribution is uniform. The form of the nonbonding
orbital may also be deduced, virtually by inspection, by rely
ing on the following device. Star the atoms in such a way as to
get the maximum number of nonneighbour stars, then the
amplitude of the orbital on each unstarred atom is zero.
Furthermore, the sum of the coefficients of orbitals on starred
atoms attached to a given unstarred atom is zero. This gives
the relative size of all the coefficients; to get their absolute
size the orbital is "normalized. The charge density on each
atom is obtained by squaring the coefficients.
The stability of evenalternants (such as 'benzene) can be
understood in terms of the preceding properties. In particular,
in an /Vatom evenalternant each atom provides one TTelectron;
each of the ^/V bonding orbitals may accommodate two
electrons ( Paul! principle), and so only the bonding orbitals
are occupied. This structural stability is further protected from
reactive attack by the uniformity of the charge distribution,
which provides no centres of attraction for potential reagents.
Questions. How is an alternant hydrocarbon distinguished
from a nonalternant? Which of the following hydrocarbons
are alternant: ethylene (ethene), butadiene (buta1, 3diene),
angular momentum
cyclobutadiene, benzene, naphthalene, anthracene, azulene,
cycfooctatetraene, phenylmethyl (the benzyl radical),
cyclopentadienyl? What properties can you predict for the
alternant hydrocarbons of this list? State the Coulson
Rushbrooke theorem. Use the form of the "benzene molecular
orbitals {p. 20) to confirm that the charge density in benzene
is uniform and in accord with the theorem. Evaluate the form
of the nonbonding molecular orbital in the phenylmethyl
radicai by the method described in note (d). (You should find
the coefficients 2A/"7 on CH 2 , and a collection of ±1//7 and
elsewhere.) Deduce from this the charge distribution in the
cation 0CH2 and the anion 0CH2. Can you deduce anything
about the chemical reactivity of phenylmethyl?
Further information. See Coulson's The shape and structure of
molecules {OCS 9), A helpful account of alternant molecules
will be found in Chapter 9 of Roberts (19616), in §26 of
Streitweiser (1961 ), in Pilar (1968), and in Salem (1966). A
book devoted to them is that of Pauncz (1967). The spectro
scopic properties are described by Murrell (1971 ) and reviewed
by Hall and Amos (1969), who furnish further directions to
the literature. The calculation of the electronic structure of
alternants has been described by Parr (1963), whose book
includes reprints of some of the original papers, by Dewar
(1969), Popie and Beveridge (1970), and Murrell and Harget
(1972). Tables of molecularorbital coefficients and energies
have been prepared by Coulson and Streitweiser (1965). For
a proof of the CoulsonRushbrooke theorem see Coulson and
Rushbrooke (1940) and a review article by Coulson (1970).
angular momentum. The angular momentum of an object
in classical mechanics is /to, where / is its moment of inertia
and oj its angular velocity (in radians per second): a big object
(with a big moment of inertia) need rotate only slowly (have
small angular velocity) in order to achieve the same angular
momentum as a small object rotating rapidly. In classical
mechanics an object may rotate with any angular momentum;
but in quantum theory the magnitude of the angular
momentum of any body is "quantized and limited to the
values {/(/+1)] !4 ri, where/ is a nonnegative (zero or positive)
integer or halfInteger (0, j, 1, . . . ). Only one component of
this angular momentum may be specified (that is, we may
state the angular momentum of a body about only one axis),
and its values are limited to mh, where m =/,/— 1, . . . — j.
This implies that, contrary to the classical situation, a rotating
body may take up only a discrete sequence of orientations
with respect to any one selected axis: the quantization of
orientation is called space quantization.
A convenient representation of the angular momentum is as
a vector of length [/(/+1)] w , see Fig. A2, which may take up a
discrete series of orientations as depicted in Fig. A3 for a body
with/=2; this is the basis of the vector model of the atom.
FIG. A2. The eiassical angular momentum and its representation by a
vector of specified projection on thezaxis.
Since only one component of this vector can be specified (and
conventionally this is taken to be the^component) the
azimuth of the vector (its orientation in the xyplane) is inde
terminate; the cone of possible orientations represents the
property of precession.
The value of the quantum number for orbital angular
momenta (the momenta arising from the spatial distribution
of the particle) is confined to integers; it is convention to use
the letter 8 to denote the orbital angular momentum quantum
number, and so perforce £ is confined to nonnegative integral
values. The intrinsic angular momentum of a particle, its 'spin,
angular momentum
1
length v^jC jH>'
classical trajectory
FIG. A3. An angular momentum with/' 2 can take only five {2/ + 1 1
orientations in space according to quantum mechanics, but all orien
tations according to classical mechanics. The discrete orientations are
illustrated.
is described by a quantum number that may. have either
integral or halfintegral values, and which is normally denoted
s (or / for nuclei).
If a system contains two sources of angular momentum its
total angular momentum is also quantized and restricted to
magnitudes [/(/+1 )] h, with/ confined to the values/[ 4/j ,
/] +/j— 1, . . . l/j— fjl, where/] and/ 2 are the quantum
numbers of the component momenta. This sequence of
numbers is known as the ClebschGordon series. As an
example, an electron with spin s — j and in an orbital with
£ = 1 constitutes a system with two sources of angular
momentum. The total angular momentum of the electron
t 1 3 1
may take on the values given by/= 1+— 1+j— 1, or ^ and ^,
depending on the relative orientation of the two momenta:
if the individual momenta are parallel the total momentum is
high (/— j), if they are opposed it is low (/—j). When the
system contains several sources of momentum the overall
angular momentum is quantized and constructed by coupling
/i and/ 2 , then/ 3 to their resultants, and so on, each step
being in accord with the ClebschGordon series.
In quantum mechanics an angular momentum can be de
fined in terms of a set of commutation rules of the appro
priate 'operators; any set of operators that satisfies the
commutation rule [/ ,/ ] = in/ is called an angular momen
tum, and the properties outlined above are common to all
such creatures. In this way the theory of angular momentum
expands to embrace intrinsic properties of systems, such as
their spin and their charge.
Questions. What are the features of quantized angular momen
tum? What magnitudes of angular momentum correspond to
quantum numbers equal to ^, 1,2, 10 M ? Draw a vector rep
resentation of an angular momentum and take £ = 1, m = 1,
0, —1; also draw the classical motion of a particle correspond
ing to the three values of m. Consider the vector representation
of the angular momentum of a bicycle wheel: is it possible to
ride a bicycle strictly perpendicular to the road? At what
velocities would this quantum wobble be intolerable? The spin
of a "photon is 1: what is the magnitude of its intrinsic angular
momentum? What angular momenta can arise from coupling
the spin of an electron with its orbital momentum in a
dorbital (£ = 2)? What states of total orbital momentum can
be obtained by coupling the momenta of two pelectrons, two
delectrons, a p and a delectron, three pelectrons?
Further information. See MQM Chapter 6 for a detailed dis
cussion of angular momentum. An interesting account which
emphasizes the connexion between classical and quantal
angular momenta is given by Kauzmann {1957). Books dealing
specifically with the quantum theory of angular momentum,
and ranging from the moderately accessible to the very
difficult, include those by Brink and Satchler {1968), Rose
{1957), Edmonds (1957), Judd (1963), and Beidenharn and
van Dam (1965); the last contains a number of important
original papers. The relation of angular momentum to the way
a system changes as it is rotated is described in MQM Chapter
6, in Tinkham (1964), which is a good introduction to the
connexion between the symmetry of a system and its angular
momentum, and in angular momentum books. The wavefunc
tion for a state of coupled angular momentum may be ex
pressed as a combination of the wavefunctions of the
contributing uncoupled states; the coefficients of the combi
nation are the "Wigner coefficients. The formal relation of
angular momentum to other properties of a system, such as
its charge, is described qualitatively by Lipkin (1965) and in
more, but not excessive, detail by Lichtenberg (1970).
anharmonicity
anharmonicity. There are two types of anharmonicity:
mechanical a n d e I ect ri ca I , Mechanics I anharmonicity (com
monly referred to simply as anharmonicity) occurs when an
oscillator is in a potential that is not purely parabolic, so that
the restoring force is not strictly proportional to the displace
ment. The energy levels in such a case are no longer strictly
those of a "harmonic oscillator, and if the nature of the anhar
monicity is to lower the potential at large displacements the
levels converge at high quantum numbers, as shown in Fig. A4.
The lines in a "vibrational spectrum in the presence of anhar
monicity are therefore no longer evenly spaced. Another effect
of mechanical anharmonicity is to ruin the "selection rules for
a harmonic oscillator: if the molecular vibration is anharmontc
it is unreasonable to expect rules developed for a harmonic
oscillator to be applicable. Therefore some forbidden tran
sitions become allowed, and harmonics of the fundamental
transitions are observed (corresponding to changes in the
oscillator quantum number by +2, +3, etc.). These tran
sitions increase in intensity with the extent of anharmonicity
in the potential.
The intensity of transitions, and the failure of the harmonic
oscillator selection rules, are also affected by the electrical
anharmonicity, which is the name applied when the dipole
moment of the molecule depends non linearly on the displace
ment. The selection rules are normally calculated on the basis
of the assumption that as the molecule is stretched the dipole
moment changes linearly with the displacement (that is, the
change in dipole moment is directly proportional to the dis
placement): if this is not so it is possible for the nonlinear
term in its true dependence, and in particular the term
quadratic in the displacement, to induce transitions by
Av = ±2, Thus electrical anharmonicity can cause intensity
changes in the vibrational spectrum of a molecule similar to
those caused by the mechanical anharmonicity; but in con
trast to the latter it does not affect the energy levels them
selves.
A further effect of anharmonicity on the intensities in a
•vibrational (infrared) spectrum of a molecule arises from its
ability to mix together vibrations of various symmetries. In
the harmonic approximation one encounters the "normal
modes of vibration: these constitute a set of independent
vibrational motions of the molecule. When anharmonicity is
present the normal modes are no longer independent, and
vibrational energy in one may leak into others. Interpreted
quantum mechanically, we say that the wavef unction for a
normal mode mixes with, and therefore acquires some of the
characteristics of , some of the other normal modes. An import
ant case in which normal modes mix as a result of anharmon
icity is Fermi resonance, which is a mechanism whereby the
simultaneous excitation of two vibrational modes (which
appears in the spectrum as a combination band) is permitted
because nearby (in energy) there is a fundamental excitation
frequency of another, allowed vibrational mode. The anhar
monicity in the molecular motion endows the mixture of
vibrational modes with some of the characteristics of the
allowed fundamental, and so the transition to the combin
ation becomes allowed. The extent to which it becomes
allowed depends on the amount of anharmonicity and the
FIG. A4. Anharmonic potentials
(shown in colour) distort the even
spacing of the levels in a harmonic
potential (black). In (a) a broader
potential reduces the separation
by different amounts; in Ibl a
narrow potential separates the
levels further; and in (c) the
complicated anharmonicity
typical of a chemical bond is
illustrated.
6 antibonding
closeness in frequency of the combination and fundamental
energies, and is greatest when they are in "resonance.
Yet another manifestation of anharmonicity is through its
effect on the moment of inertia of a molecule, and through
that on the molecule's "rotational motion and spectrum. A
harmonically vibrating molecule has the same mean size what
ever its vibrational state; but an anharmonic molecule would
tend to swell slightly, and to change its moment of inertia, as
it is excited to higher vibrational states (see Fig. A4c). The
dependence of the moment of inertia on the vibrational state
affects the structure of the "branches in the vibration rotation
spectrum.
Questions, 1 . What types of anharmonicity exist, and what do
they affect? Discuss the effect of replacing a harmonic oscil
lator potential by one that is almost parabolic, but (a) gets
broader, (b) gets narrower as the displacement increases. What
is expected to be the form of the anharmonicity for a typical
bond stretch? Discuss the vibrational potential for the outof
plane vibration of a planar molecule, with special reference to
the anharmonicity. What previously forbidden transitions
become allowed in the presence of anharmonicity? What tran
sitions does electrical anharmonicity permit? What effect does
electrical anharmonicity have on the energy of an oscillator?
What effect does the presence of anharmonicity have on the
symmetry selection rules? What is the grouptheoretical inter
pretation of this? What is a combination band, and why does
Fermi resonance endow it with intensity? What effect is there
on the intensity of the allowed fundamental when it takes part
in Fermi resonance? What happens to the latter's energy? What
group theoretical reason accounts for our stressing the role of
a combination band in Fermi resonance rather than simply
another fundamental? (Consider the symmetry of the anhar
monic part of the molecular energy .J
2. Consider a potential of the form ^kx 2 + ax. Sketch the
form of the potential on the assumption that a is small, and
apply secondorder "perturbation theory to the calculation of
the effect of the linear anharmonicity on the potential. Show
that the electric dipole moment may depend both linearly and
quadratically on the displacement of the molecule from
equilibrium (use a Taylor expansion) and that the quadratic
term can induce transitions disallowed in a harmonic oscillator.
Use the properties of the harmonic oscillator in Table 1 1 on
p. 273.
Further information. See MQM Chapter 10 for a discussion of
anharmonicity, and a discussion of the role of symmetry in
governing what vibrations Fermi resonance may mix together.
Woodward (1972) has a helpful qualitative discussion in §185
and 22*10; so too do Brand and Speakman (1960) in §6*7
and King (1964) in §55. See also Barrow (1962), Whiffen
(1972), Gans (1971), and Wilson, Decius, and Cross (1955).
Extreme anharmonicity leads to dissociation: the extrapolation
of anharmonicity to this limit is discussed in Chapter 5 of
Gaydon (1968).
antibonding. An antibonding orbital is one that, when occu
pied, tends to induce dissociation. Imagine the Isatomicorbitals
on two "hydrogen atoms which are being brought together,
and suppose that the signs of the amplitudes of the two wave
functions are opposite. The effect of bringing the atoms
towards each other is to slide the region of positive amplitude
of one wave into the region of negative amplitude of the other,
and, just as in the case of conventional wave phenomena, the
waves interfere destructively and the total wave amplitude in
the region of overlap is diminished. The square of this ampli
tude determines the probability of finding the electrons in a
particular region, and so the effect of bringing the orbital s
together with opposite phases (signs) is to diminish the
electron density in the internuclear region. This has an adverse
effect on the energy of the molecule (because the internuclear
region is the best place to put the electrons, for then their
interaction with the two nuclei is the most favourable), and
the molecule formed in this way will have an energy greater
than that of the two hydrogen atoms at infinite separation:
this is therefore a dissociative situation, and the molecule is
unstable. The orbital responsible for this instability is referred
to as an antibonding orbital.
The case of two helium atoms being brought together is a
good example of the effect of antibonding character. When
the two nuclei are quite close together the Isorbitals overlap
appreciably: the amplitudes taken with the same phase inter
fere constructively in the internuclear region to give one
"molecular orbital, and the amplitudes taken with opposite
antisymmetry
sign interfere destructively to give another, antibonding,
molecular orbital. Four electrons have to be added to these
two composite molecular orbitals: two enter the lowerenergy
orbital and tend to cause the nuclei to stick together; the next
electron enters the antibonding orbital and so lowers the
strength of the bond. The fourth "pairs with the third in the
antibonding orbital, and the combined effect of this pair is
sufficient to overcome the bonding of the first pair and to
disrupt the molecule. Consequently He 2 is an unstable mole
cular species (even though HeJ is weakly stable in the gas
phase). This description enables one to see why two helium
atoms collide without sticking together: as the atoms approach
the bond and antibond are formed, but both are occupied
simultaneously and the rise in energy as the atoms approach
appears as a repulsive force. It is quite easy to extend this de
scription to more complicated atoms and molecules and to
understand why bulk matter is impenetrable.
Questions. 1 . What effect does the occupation of an antibond
ing molecular orbital have on the energy of a molecule? Why
does the overlap of atomic orbitals with opposite sign lead to
an antibonding orbital? Why is HeJ less stable than He* ( and
He 2 unstable? Why, then, is the world not littered with He*?
Why do not two neon atoms stick together when they collide
in a gas, and how may this argument be adapted to account
for the same property of methane and N 2 ? Why is bulk matter
impenetrable?
2, Take a Isorbital on each of two protons at a separation R,
and use the mathematical form for the orbitals given on p. 275
to plot the amplitude of the molecular orbitals that result
when they are combined first with the same sign and then with
opposite sign. Plot the electron density corresponding to two
electrons in the bonding orbital and then to two in the anti
bonding orbital, and then plot the difference density (obtained
by ignoring interference effects, calculating the electron
density when each electron is confined to its own nucleus, and
subtracting this density from the density calculated for the
bonding and antibonding cases). Do this calculation for about
three judiciously chosen nuclear separations, and reflect on
the connexion of these results with the discussion in the text.
A proper calculation should use "normalized orbitals, but a
simple one is sufficient for illustration. What effect does
normalization have on the difference densities?
Further information. Antibonding effects are of considerable
importance in determining molecular structure: see Coulson's
The shape and structure of molecules (OCS 9) and Coulson
(1961). A long essay on the importance of antibonding
orbitals has been written by Orchin and Jaffe (1967). Further
details will be found in MQM Chapter 9, and helpful advice on
the calculation of overlap integrals is given by McGlynn,
Vanquickenborne, Kinoshita, and Carroll (1972) Chapter 2 and
Appendix C. A compilation of molecular energies has been
published by Richards, Walker, and Hinkley (1971). Difference
density maps are given by Coulson (OCS 9) and by Deb
(1973), and the latter gives references to many other sources.
antisymmetry. A function f (x) is antisymmetrical (or
antisymmetric) if f{— x) is equal to — fix). A wavefunction of a
system containing N indistinguishable particles h antisym
metrical under particle interchange if it changes sign when the
coordinates of any pair of particles are interchanged; that is, if
^(ri,r 2 ) = ~"lM r J< r i ) Do not confuse the word with
asymmetrical, which means the absence of symmetry. An
example of the first type of antisymmetry is the function x,
because (— x) = — (x); another example is the function sinx,
because sin(— x) = —sin x. An example of an antisymmetrical
wavefunction is i/' a (i'i)0' b (r 2 ) — (r 2 ) '/' b (r 1 ) because inter
changing the coordinates rj and r 2 , which by reference to
Fig. A5 is seen to be equivalent to interchanging the particles,
changes the sign of the function.
FIG. A5 Interchanging particles 1 and 2 is equivalent to interchanging
the vectors t\ and rj.
8
aromaticity
The importance of antisym metrical wavefu net ions stems
from the "Pauli principle, which demands that the total wave
function of any collection of electrons must be antisym
metrical. The formation of a "Slater determinant is one way
of constructing a fully antisymmetrized function out of a
collection of functions. The imposition of the requirement of
antisymmetry on a wavefunction introduces important modi
fications to the energy: see "exchange energy.
Questions. State the condition on f{x) for it to be an antisym
metrical function of x. Which of the following functions are
antisymmetrically 2 , x 3 , 3x 2 — 2x 3 , cosx, tan x, expx, expx 2 ,
cosecx? Show that any asymmetric function F(x) can be
expressed as the sum of an anti symmetrica I function and a
symmetrical function. What is meant by an antisymmetrical
wavefunction? Which of the following wavefu net tons are
antisymmetrical: \p (r ( )i£ (r a ), sin [*{ri~r 2 )],
*.(ri »* b ('a )* c ('3»  * B (r 2 )^ b (r, )^ e (r 3 ) +
^ a (r 2 )^ b (r 3 f ^(rj ) — ...? Show that the last can be
written as a 3 X 3 determinant. What is the importance of
antisymmetry in quantum mechanics?
Further information. See MQM Chapter 8 for the reasons lying
behind the use of antisymmetric wavefuncttons. Helpful dis
cussions of the reasons and the consequences will be found in
§29 of Pauling and Wilson (1S35), in §31 of Slater (1963),
and §T2 and succeeding chapters of Richards and Horsley
(1970). The symmetry and antisymmetry of functions can be
envisaged, and treated, as a problem for "group theory; there
fore see MQM Chapter 5, Cotton (1963), Tinkham (1964),
Bishop (1973), and Altmann (1962). The antisymmetrization
of wavefunctions is a problem in statistics: see Pauling and
Wilson (1935), Condon and Shortley (1963), Hamermesh
(1962), and Judd (1963). This reading sequence will illustrate
how a simple requirement may have consequences of un
bounded complexity.
aromaticity. An aromatic molecule is cyclic, planar, and con
jugated (possessing alternating single and double bonds) but
with a stability greater than would be expected for a molecule
with so many double bonds. The extra stabilization is due to
■resonance (in "valencebond language) or derealization (in
"molecularorbital language), and the extra stabilization energy
is called the "resonance energy or derealization energy.
Benzene is the archetype of such molecules. It has been found
that aromatic molecules possess 4/J+2 electrons (the Hiicket
4n+2rule), where n is an integer. Thus benzene has n = 1, and
the simplest aromatic molecule of all, which was first prepared
not long ago, is the cyclopropene cation (/? = 0). The large
molecule [18] annulene, consisting of 18 conjugated carbon
atoms in a planar ring, is also aromatic {n = 4),
The basis>of the An+2 rule may be understood by con
sidering the energy levels of cyclic hydrocarbons. From N
atoms N molecular orbitals may be formed. The lowest
energy (most strongly binding) orbital has no "nodes, all the
others have nodes, and, except perhaps the uppermost, are
doubly "degenerate. See, for instance. Fig. A1 on p. 2, (This
twofoid degeneracy may be viewed as a consequence of the
fact that an electron may run round the ring in either
direction.) In the case of N being an even number the upper
most level is nondegenerate. Into the lowest energy level one
may insert two electrons, and into each doubly degenerate
pair one may insert four electrons. Closedshell molecules
therefore contain 4/J+2 electrons, where n is the number of
fully occupied doubly degenerate levels. Achieving a closed
shell is an energetically favourable situation, especially as one
is normally obtained by adding no more electrons than are
needed to complete the bonding molecular orbitals, and so
leaving empty the energetically unfavourable "antibonding
orbitals.
It has also been found that some molecules of a conjugated
doublebond nature show an enhanced instability: they
contain 4n electrons, and include the cyclopropene anion
(n = 1 ). This great instability gives rise to the name
antiaromatic.
Further information. See MQM Chapter 10, Coulson's The
shape and structure of molecules (OCS 9), Coulson (1961 ),
Streitweiser (1961), Salem (1966), and Pilar (1968). A simple
account of the preparation and properties of a variety of
aromatic and antiaromatic molecules has been given by
Breslow (1972). The analysis of the 4n+2 rule may be carried
fbrther by referring to Streitweiser (1961), Chapter 10; there
will be found described the concept of pseudoaromaticity and
Craig's rules. See Bergmann and Pullman (1971 ) for the pro
ceedings of a conference on aromaticity, pseudoaromaticity,
and antiaromaticity,
atomic orbital. An atomic orbital describes the distribution
of an electron in an atom: it is the °wavef unction for an
electron in an atom. The classical Rutherford and °Bohr
theory of the hydrogen atom sought a model of its structure
in terms of a trajectory of the electron about the nucleus, and
so the atom was viewed as a central nucleus with an electron
in one of a variety of orbits. The introduction of quantum
mechanics, and in particular the impact of the "uncertainty
principle, showed that the concept of trajectory was untenable
on an atomic scale, and so an orbit could not be specified.
Quantum mechanics replaced the precise trajectory, the orbit,
of the electron by a distribution, an orbital. An atomic orbital
is a function i^(r} of the coordinates of the electron, and, in
accord with the Born interpretation of the 'wavef unction, the
probability that the electron may be found in an infinitesimal
volume element dr surrounding the point r is \p*{r)\p{r)<iT. It
follows that, if the form of the atomic orbital is known, we
are able to predict the electron density at any point in the
atom.
As an example, the electron in the ground state of the
hydrogen atom is distributed in an atomic orbital (an sorbitat)
of the form exp (— r/a ), where a is a constant, the Bohr
radius (529 X 10~ n m): this implies that the orbital is
spherical (the function depends on r but not on or 4>). and
the electron density at any point depends on exp ( 2r/a ).
Therefore the density is greatest at the nucleus, and then
declines exponentially with distance. Such an atomic orbital
can be represented by a spherical boundary surface within
which there is some probability, let us say 90 per cent, of
discovering the electron. In this boundary surf ace represen
tation, which is depicted in Fig. A6, the sorbital is drawn as a
sphere of an appropriate radius; but it must not be forgotten
that there is some slight probability of discovering the electron
at points well outside the boundary surface.
It is normally sufficient in the discussion of atoms to con
fine attention to the s, p, d r and occasionally the forbitals.
Of these only the sorbital is spherically symmetrical; the
others have an increasingly pronounced angular dependence
corresponding to the electron being concentrated in particular
atomic orbital 9
directions in space. The form of these atomic orbitals will be
found in the discussion of the "hydrogen atom.
The amplitude of an atomic orbital depends on the distance
from the nucleus. It is reasonable to expect the amplitude to
diminish to zero at large distances from the parent nucleus,
and this is found to be so: when r is very large all orbitals
decay exponentially. The decay is not in general exponential
at all distances from the nucleus, for in most some incipient
undulations occur at small radii. This behaviour is examined in
detail in the case of the "hydrogen atom, and only one point
need be emphasized here: only for sorbitals does the
amplitude not drop to zero at the nucleus itself (see Fig. A6).
FIG. A6. Boundary surface capturing different proportions of the
electron in the ground state of the hydrogen atom (in a 1 sorbital), R
denotes the covalent radius (30 pml.
The number of lobes possessed by an atomic orbital
(actually the number of angular modes) determines the
"orbital angular momentum of an electron in the orbital. The
mean curvature of the orbital, which is determined by the
number of radial and angular nodes (because the more often
the wave must pass through zero, the more sharply is it curved),
determines the "kinetic energy of an electron that occupies It,
and the mean inverse distance from the nucleus determines the
potential energy in hydrogen and the attractive part of the
potential energy in manyelectron atoms. In manyelectron
atoms the energy is also influenced by the interelectronic
repulsions, and these have the further effect of distorting the
electron distributions from those in hydrogentike atoms; these
10
atomic spectra: a synopsis
effects are calculated by "self consistent field methods.
Although true orbitals of manyelectron atoms are complicated
functions, it is possible to make a fair approximation to them
by orbitals that have the same angular dependence as those of
hydrogen, but whose radial dependence is determined by a set
of simple rules: these are the "Slater orbitals.
When atomic orbitals are represented by boundary surfaces
it must be borne in mind that in manyelectron atoms these
are only a crude representation of the actual electron dis
tribution, and in fact do little more than designate regions of
space where the orbital has appreciable amplitude and where
the electron has a high probability of being found.
Questions. What is an atomic orbital? What information does
it contain about the distribution and properties of an electron
that occupies it? How does an orbital differ from an orbit?
What is meant by a boundary surface? Calculate from the
wavef unction given in the text the radius of the boundary
sphere which captures 50 per cent, 80 per cent, 90 per cent,
and 9999 per cent of the electron density in the Isorbital of
hydrogen (a = 53 pm, 053 A), What is the effect on the
shape of atomic orbitals of the interelectronic repulsions in
many electron atoms, and how may they be taken into
account? Sketch the boundary surfaces for p, d, and
forbitals by referring to the pictures in the article on the
•hydrogen atom. What is the evidence that electrons are dis
tributed in atoms In the manner we have described?
Further information. See MQM Chapter 8. The shape and
significance of atomic orbitals are discussed in detail in
Coulson (1961), Herzberg (1944), White (1934), Pauling and
Wilson (1935), and Kauzmann (1957). Information about
selfconsistent field and Slater orbitals wili be found under
the appropriate headings. The electronic occupation of atomic
orbitals is determined by the 'aufbau process and the "Paul!
principle. As well as determining the structure of atoms the
atomic orbitals are the basis of descriptions of molecular
structure: see "molecular orbitals and linear combination of
atomic orbitats. See also "wavef unction and "radialdistribution
function. Read the section on the hydrogen atom for a
detailed discussion of its orbitals.
atomic spectra: a synopsis. An atomic absorption or
emission spectrum arises when an atom makes a transition
between two states (which are often called "terms): the
combination principle states that all lines in a spectrum can
be represented as the difference between two terms {for the
word 'term' is also used to denote the energy of a term). The
transitions observed can normally be ascribed to "electric
dipole transitions, and the lines that may appear in the
spectrum are governed by the "selection rules. Their intensity
is determined by the magnitude of the transition dipole
moment. The appearance of the spectrum may be modified
by the application of a strong magnetic field (the "Zeeman
effect) or a strong electric field (the "Stark effect), because
both fields can cause small shifts in the energy levels of the
atom. On the gross structure of the spectrum is seen a "fine
structure, which is interpreted in terms of spinorbit coupling,
and an even finer "hyperfine structure which is due to the
interaction of the electrons with the "magnetic dipole and
electric "quadrupole moments of the nucleus.
The information of chemical interest that can be obtained,
or has been obtained, from the spectrum of an atom is as
follows:
1. Identification of species. Since every element has a
characteristic spectrum, atomic spectroscopy m3y be used in
analysis: the spectrum is used as a fingerprint for elements.
2. Evidence for quantization. The study of the spectrum of
atomic "hydrogen was of profound significance for developing
the ideas of "quantum theory and quantization. Out of the
study emerged the idea of "atomic orbitals, and all the other
paraphernalia of quantum chemistry.
3. The Pau/i principle. From a study of the spectrum of
helium emerged the puzzling result that not all the states of
the atom are allowed. The "Pauli principle was the rational
ization of these data, and its discovery was the key that
enabled the periodic system to be explained.
4. Atomic energy levels. The study of atomic spectra yields
information on the energy levels of atoms: we are able to say
how deeply electrons are buried in inner shells and which
electrons and states of the atom are likely to be important in
atomic units
11
governing the bonding properties of atoms {their valence, and
the strength of the *bonds they form). We need to know the
energy of atomic energy levels in order to assess the energy of
the "valence state and the rote of "hybridization. Photo
chemistry depends on a knowledge of the energy levels of
excited atoms.
5, Ionization potentials. The energy required to ionize an
atom (its "ionization potential) can be determined from atomic
spectroscopy; so too, with more difficulty, can some "electron
affinities. Both these properties are central to an understanding
of the structure and reactions of atoms and molecules. See, for
example, "electronegativity.
6. Spinorbit coupling. From the fine structure we may
determine the spinorbit coupling constant. This is of use in
the discussion of the role of triplet states in photochemistry
(for example, the heavy atom effect in quenching "phos
phorescence) because it determines the rate of singlettriplet
intersystem crossing. We also need to know spinorbit
coupling constants to evaluate ^values in "electron spin
resonance and to discuss the structure of molecules— see, for
example, the Hund coupling cases.
1 . Hyperfine coupling constants. From the "hyperfine
structure of spectra can be determined the strength of the
magnetic and electric coupling of electrons to nuclei, and also
the °spin of nuclei. Such coupling constants are important for
the "hyperfine effect in "electron spin resonance and the
"spinspin coupling in "nuclear magnetic resonance.
8, Xray Spectra, Spectra in the shortwavelength °Xray
region were the basis for Moseley's determination of the
atomic numbers of the elements.
Further information. See MQM Chapter8foradescriptionof
the interactions that lead to the structure of atomic spectra
and a more detailed description of their form. Introductory
books on atomic spectra include those by Whiffen (1972),
Barrow (1962), Herzberg (1944), Woodgate (1970), and
White (1934). A book with many examples and with compre
hensive coverage at a slightly more advanced level is that by
Kuhn (1962). More detailed analysis will be found in Shore
and Menzel (1968), Candler (1964), who gives much exper
imental data, Condon and Shortley (1963), Judd {1963,
1967), and Wigner (19S9). The books by Condon and
Shortley and Wigner are classics: the former was written
before many of the impressive angularmomentum techniques
were developed but has been a dominating influence on the
development of the subject, and the tatter is a classic and
original exposition of atomic structure and spectra in terms of
symmetry and "group theory. Both books may be regarded as
ancestors of Griffith (1964), who, after his description of free
atom spectra, develops the theory of the spectra of atoms in
complexes: see "crystal fie Id theory. Data from atomic
spectroscopy will be found in Moore (1949 et seq.), who lists
energy levels. Applications to photochemistry are described by
Wayne (1970) and Calvert and Pitts (1966).
atomic units. The appearance of many equations in quantum
mechanics may be considerably simplified if mass is expressed
as a multiple of the electron mass m (so that the mass of
the electron is taken to be unity); charge as a multiple of
the proton's charge e; length as a multiple of the 'Bohr
radius a ; and energy in multiples of twice the ionization
potential of the ground state "hydrogen atom. {Twice the
ionization potential, 2721 eV, ore 2 /47re a , is generally
employed, although some people use the energy itself; that is,
1365 eV.) A consequence of this choice of units is that h= 1.
The units may be augmented by the choice c = 1 for the speed
of light (and so all velocities are expressed as a fraction of the
speed of light in a vacuum). The units so chosen eliminate
many of the constants in the "Schrodinger equation, and the
numbers that emerge for various properties can be translated
BOX 1: Atomic units (a.u.)
quantity value
1 a.u. of
mass m a 9109 X ID" 31 kg
length a ° 5292 X 10" n m
charge e 1602 X 10" 19 C
energy e 2 /4m g a 272 eV; 2625 kJ mof '
velocity c 2998 X 10 8 m s" 1
Consequently ft = 1 ; $i B = 5; ff H = j.
12
aufbau principle
into conventional units by reintroducing the units of mass,
length, charge, and energy; see Box 1.
aufbau principle. The aufbau or buildingup principle is the
statement about how electrons should be fed into the orbitals
of an atom or molecule in order to construct the species. The
principle states that an electron enters the lowest available
orbital consistent with the requirements of the "Pauli exclusion
principle. This implies that the first electron enters the lowest
orbital, the second joins the first (but with opposite "spin);
the third electron enters the next higher orbital, then the
fourth pairs with it and so on. See Fig. A7, When a set of
"degenerate orbitals is to be filled {for example, when the p
shell of an atom is being populated) the first electrons enter
different members of the set with parallel spin, in accord with
the °Hund rule, and only when each degenerate orbital con
tains one electron do the remainder enter with paired spins.
The list of the orbitals populated by the application of the
aufbau process constitutes the "configuration of the atom.
FIG. A7. The order of filling energy levels according to the aufbau
principle.
Questions, What is the aufbau principle? In what way does it
depend on the Pauli principle? How would the principle
differ if the particles being filled into the orbitals were
"bosons? What is the role of Hund's rule? Discuss the structure
of the atoms He, Li, Be, B, C, and N in the light of th^ aufbau
principle, and also the structure of the molecules H 2 , N s , 2 ,
Fj, and NO (get help from "molecular orbitals). Discuss what
might happen when two energy levels lie close together, but
are not degenerate: under what circumstances might each be
halffilled before the lower is filled?
Further information. See MQM Chapter 8, Chapter 3 of
Herzberg (1944), and Murrell, Kettle, and Tedder (1965).
The aufbau principle is the basis of the periodic table; there
fore see how it is applied in Puddephatt's The periodic table of
the elements (OCS 3) and in Chapter 2 of Phillips and Williams
(1965). For the application of the principle to molecular
systems see MQM Chapter 9, Coulson's The shape and
structure of molecules {OCS 9), and Coulson (1961 ). The
aufbau principle is important in transitionmetal chemistry
because in complexes the metal ion has a number of closelying
energy levels, and the situation in the last question is common:
see "crystal fie Id theory and "ligandfield theory.
Auger effect. Other names for this effect are autoionization
and preionization. The Auger effect is a radiationiess tran
sition (a transition between states that involves no emission or
absorption of radiation) from an excited state into a dis
sociative state. Consider an atomic energy level scheme of the
type shown in Fig. AS: in this atom one series of levels
terminates {at an ionization limit) at a lower energy than the
other. Suppose we monitor the absorption spectrum of the
atom, and we concentrate on the series on the left (l_>. The
spectrum observed consists of a series of sharp lines at increas
ing frequencies, but when the frequency corresponds to an
energy above the ionization limit of the other series (R), for
example when we observe the transition to the line A, a
marked change appears in the spectrum. The most noticable
difference is in the sharpness of the lines, for A,B ... are
blurred. They may also be slightly shifted. What is happening
is that some "perturbation in the atom (for example, the spin
orbit coupling) is mixing the states of series L with those of
series R, and therefore the states of L take a little of the
character of R. But above the ionization limit of R this implies
that the states of L have a tendency to ionize, the tendency
Auger effect
13
\orta&tQ$ limit of I
rodiotjanfcu
transitions
i8sia3tian(ni.of, R
spectrum end harwtions of L
FIG. A8. The Auger effect: the broadening of the spectrum (on the left)
occurs where the atom makes radiationless transitions from states on the
teft to the unbound states on the right.
increasing as the contamination increases. Since the lifetime of
the levels A, B, ... is diminished by this mixing, the width of
the levels is increased ("uncertainty principle). The mixing of
states R into states L may be expressed as a probability that an
atom in a state of L makes an actual transition into a state of
R, and the Auger effect is simply this type of radiationless
transition into a dissociative state. The name preionization
reflects the fact that ionization occurs in series L before, on
energetic grounds, it is expected, and the name auto ionization
reflects the 'selfinduced' nature of the process in the sense that
the perturbations within the molecule induce the ionization by
flipping the bound state into the ionizing state.
The Auger effect was originally detected in "Xray
spectroscopy, where the bombardment of a solid with fast
electrons excites a Ksheli electron (let us say), and an Xray
is emitted when an Lshell electron falls into the vacant hole.
A competing process is introduced by the Auger effect,
because the excitation of the Kelectron may induce auto
ionization, and another electron is boiled off the atom or out
of the solid. The ionization process competes with the forma
tion of Xrays and diminishes their intensities. The effect is
not wholly bad for, if the energy of the Auger electron is
measured, information may be obtained about the energy
levels of electrons in solids. This is the basis of Auger spec
troscopy.
For the sake of completeness. Auger is pronounced 036.
Questions. What is the Auger effect? How may it be detected
in atomic spectra? Why do spectral lines become broadened by
virtue of the Auger effect? What perturbations may cause auto
ionization? Describe the appearance of the spectrum showing
preionization. What is the role of the Auger effect in Xray
spectra? What effect does it have on the lines? Suppose we
were looking at an atomic emission spectrum, what would be
the influence of the Auger effect? In what sense may "predis
sociation of molecules be considered to be an Auger effect?
What other processes can you think of that, however loosely,
may be considered to be the manifestation of an Auger effect?
Further information. An account of the Auger effect in atomic
spectra will be found in §4.2 of Herzberg {1944) and §4.5 and
§5.3 of Kuhn (1962). The rote it plays in the formation and
appearance of Xray spectra is described by Burhop (1952)
and has been reviewed by Burhop and Asaad (1972). For a
discussion of Auger spectroscopy, see Siegbahn (1973) and
further references under "ionization potential. See also
"predissociation.
B
band theory of metals. The theoretical description of the
structure and properties of metals is based on the view that in
gross terms they are composed of an array of positive ions held
together and surrounded by a sea of electrons. The properties
of this sea of electrons determine the typical characteristics of
metals: their electrical conductivity, thermal conductivity,
reflectivity, malleability, and ductility. The energy levels
available to the electrons are of paramount importance, and
forbidden
enemies
permitted
energies
4
forbidden
eneiqies
permitted
eneiqie*
T
r mm
FIG. B1 . The extent of occupation of bands separated by gaps deter
mines whether a material is a conductor or an insulator. Occupied levels
are coloured.
analysis of the problem shows that the available energies fall
into bands, and that between these bands lie regions of energy
which no electron can possess. If a band is less than full
(Figs. B1a and c), the electrons in it can be induced to move
under the influence of a small disturbance: hence the high
conductivity of metals. If the bands are full, electrons can be
induced to move only if they can be supplied with enough
energy to excite them through the forbidden band gap into
an empty upper band: this is an energetically unfavourable
situation, and so such materials are electrical insulators
(Fig. B1b).
The formation of energy bands in metals may be described
in a number of ways: we shall discuss two. The first starts
from the view that a metal may be envisaged as a massive
molecule, a molecule of almost infinite extent. The molecule
is composed of atoms in a regular lattice, and on these atoms
there are "atomic orbitals which overlap with their neighbours'
atomic orbitals. This situation of overlapping orbitals is
encountered on a much smaller scale in conventional molecules,
for it is the basis of the "molecularorbital description of mole
cular structure. Let us pretend that only a single sorbital is
available on each atom, and for simplicity we consider a linear
crystal (a single line of N atoms}.
Concentrate first on two of the atoms in the chain: their
orbitals 'overlap and form a bonding and an antibonding pair
of molecular orbitals with an energy separation determined
largely by the amount of overlap (Fig. 82], Now bring up a
third atom to a lattice distance and let it overlap with its
immediate neighbour (and for simplicity, and as a good ap
proximation, neglect its small overlap with its next nearest
14
band theory of metals
15
c
m
N~oo
FIG. 82, As 2, 3, 4, ... N atoms are added in a line the energy levels
begin to fill a band; when N is very large the energy levels with in the
band virtually form a continuum.
neighbour). This adjusts the molecular orbitals and their
energies so that three orbitals are obtained in the place of the
former two; one is bonding, one is nonbonding r and the third
is antibonding. This is illustrated in Fig. B2. The process of
sliding an atom along the line up to a lattice point [rather like
using an abacus) may be continued, and Figs. B2 and B3
show that the effect is gradually to fill in the energies until
when N is very large the N energy levels form a virtually con
tinuous band of energies of width A , Since the band has been
formed from sorbitals it is called the sband.
If each atom contributes one electron (the atoms might be
sodium atoms in a line), then since each energy level consti
tuting the band may accommodate two electrons ("Pauli
principle), in the metal half the band will be full of electrons,
and the upper half will be empty, as in Fig. BTa. (This is just
the application of the °aufbau process to a collection of energy
levels.) The presence of the halffilled band means that the line
of atoms behaves as a metal. If instead each atom were to con
tribute two electrons, the band would be filled, as in Fig. B1b,
and an applied electric field would be unable to shift the
electrons: the line of atoms is then an insulator. Why motion
cannot occur in this case will be described below in more
detail.
It is improper to restrict the formation of a band to the
overlapping of sorbitals. If the metal atoms have porbitals in
their valence (outermost) shell these too should be allowed to
overlap. Because of shielding and penetration effect, porbitals
lie higher in energy than sorbitals, and so their overlap gives
rise to a band (the pbancl) above the sband, and separated
from it by a gap, the magnitude of which depends on the
strength of bonding between the atoms and the sp separation
of the isolated atoms (Fig. B4). Here we see the reason for the
appearance of the bands of energies, and the regions of
forbidden energies.
The approach to the band structure just described is some
times referred to as the tightbinding approximation (TBA)
because it takes the view that electrons tend to stick to nuclei,
and therefore are quite well described by atomic orbitals
characteristic of the atoms. An alternative approach starts
from the view that a good first approximation is to suppose
that the electrons have no electrostatic interactions and are
completely free to swim about in the bulk of the metal, and
that the presence of a periodic lattice may be imposed as an
improvement in a second step: this is the nearly free electron
approximation (NFE).
How does this alternative approach lead to band formation?
Let us once again consider the onedimensional metal, but
begin by ignoring the presence of positive ions. The situation is
now that of a collection of 'particles in a box, or, since the
line is of almost infinite length, the same as the freeelectron
system. The energies available to free electrons take a con
tinuous range of values and depend on the momentum ±kh
according to k 7 (Fig. B5a). Pouring electrons into this con
tinuum of energy levels fills them up to some energy E , the
Fermi energy, and leaves vacant all the energy levels above.
The filled levels constitute the Fermi sea, whose surface, the
Fermi surface, is at E p . On this model everything would be a
16 band theory of metals
Enenjy
FIG. 63. The actual snergv levels
for a chain of N atoms, each inter
acting only with its nearest
neighbour. Note that by N  20
the band is getting dense, and that
its width does not expand indefi
nitely.
6 7 8 9
Number of otorrn in row
II 12 20
! pband
sp separation in oroms
band qap in metal
FIG. B4. Bandgap formation, s and pbands, and the orbital overlaps
corresponding to the extreme energies of each band.
metal, for the 'band' has no upper limit, and so can never be
filled. This emphasizes the central role that the periodic lattice
plays in determining whether or not a material is a metal.
In order to see how the lattice imposes a band structure we
remind ourselves that the states of the free electron are waves
of definite length (wavelength = 2ir/k), and the true system
may be regarded in terms of these waves propagating through
the periodic lattice. At wavelengths long compared with the
lattice spacing a the waves slide through, but when the wave
length is comparable to a the lattice diffracts the waves. When
the wavelength is equal to twice the lattice spacing a pattern of
standing waves is set up because a wave that begins moving to
the right is reflected by the lattice and moves to the left,
where it is reflected to the right, and so on. These standing
waves, with wavelengths in the vicinity of 2a, have a stationary
distribution in space, and we may envisage two types of
arrangement. In one arrangement the standing wave has its
amplitude maxima at the positions of the lattice points
(positive ions), and in the other the maxima are between the
lattice points. Whereas in the freeelectron model these stand
band theory of metals
17
second
Briiiouin lone
first
BfffJouin zone
FIG. B5. (al The freeelectron energy levels and (b) the band structure
imposed by the periodic lattice. Both contain a number of electrons
that makes them metals.
ing waves had the same energy, the presence of the periodic
lattice implies that they have different potential energies, and
so we discover that at k = ±ir/a there is an energy gap (see
Fig. B6). Another gap is found at k = ±2n/a, and so on; there
fore, we see that the periodic lattice splits the freeelectron
continuum of energy levels into a series of bands separated by
gaps (Fig. 65). Into these bands, the Briiiouin zones, may be
inserted the electrons of the metal, and metallic properties
are predicted only if there remain incompletely filled bands.
The relation of the NFE bands to the tightbinding bands
may be discovered by comparing the form of the orbitals at
the edge of the s and pbands with the waves at the edges of
the first and second Briiiouin zones. The immediate obser
vation, from Fig. B6, is that the noda! structure is the same;
only the details of the electron distributions differ, especially
in the regions close to the nuclei.
Well below the Briiiouin zone boundary (that is, for energy
levels for which \k I < ir/a in the first zone) the states of the
electron may be regarded as running waves {as opposed to the
standing waves forced on us at the zone edge). In one dimen
sion the waves run with momentum +kh to the right and —kh
to the left; since they have energy proportional to k 2 these
two running waves are "degenerate (have the same energy).
FIG. B6. Energygap formation due to a periodic lattice and its relation
to the orbital (TBA) approach (in colour). Note that the uppermost
level of the lower band and the lowermost of the upper have the same
wavelength but one has nodes at the nuclei, the other between them.
The lattice spacing is a.
18
band theory of metals
Therefore, in a metal there are equal numbers of electrons
running to the left and to the right, and in the absence of
external stimuli no current flows. When an electric potential
difference is applied, the energy of the electrons running to the
right may differ from those running to the left (Fig, B7) and
so the electrons redistribute themselves in order to attain the
lowest energy: this means that more occupy states corre
sponding to the flow of electrons to the right, because a lower
energy is attained thereby, and the equilibrium situation shows
a steady flow of current. If however the band is full the elec
trons cannot reorganize themselves to give a net direction to
travel (consider Fig, B7 with a full band), and so no current
flows. Such a material is an insulator. A really enormous field
might so affect the energies that an empty zone is brought
down to the filled one: in this case of dielectric breakdown a
current may flow, but often at a cost of disrupting the
material.
k— ft/a
FIG. B7. When a suitable potential difference is applied to a material
with an unfilled band the lowest energy distribution of electrons has
more moving to the right [positive k) than to the left (negative k).
The band theory of metals is able to account for the
thermal properties of metals. For example, the high thermal
conductivity may be traced to the way in which an electron
carries heat rapidly through the lattice. Applying heat to one
end of a metal rod induces lattice vibrations: these lattice
vibrations excite an electron with an energy close to the Fermi
surface into an unfilled state just above the sea, and so it may
skim through the lattice with its high energy. Sooner or later it
will plunge back into the Fermi sea, and in the process will
impart its excess energy to the lattice, but this part of the
lattice might be some way from the heated end of the rod.
Thus heat is conducted through the rod.
Pioneers of the theory of heat capacities of metals were
worried by the presence of a great sea of electrons, all of which
ought to be able to contribute to the heat capacity of a block
of metal. Band theory dispelled this worry by drawing atten
tion to the fact that when a metal is excited thermally an
energy of the order of kT\s supplied, and only those electrons
within an energy kT of the Fermi surface can be excited into
an empty level by such a stimulus. Consequently only a very
small number of electrons are able to contribute to the heat
capacity of a metal.
Questions. 1. A metal may be pictured as an array of positive
ions embedded in a sea of electrons: how is this picture able to
account for the malleability and ductility of metals? What
other characteristics of metals must any theory of their struc
ture explain? What is the role of energy bands in determining
electrical and thermal conductivity? Why are clean, smooth
metal surfaces highly reflecting? Describe the tightbinding
approximation, and explain why bands may be formed. What
is the difference between s and pbands? Consider a one
dimensional metal lattice, and let there be s and porbitals in
the valence shell. Describe the number of electrons that would
give rise to metallic or insulating properties to the system
(consider only porbitals along the tine). What determines the
width of the bands? Guess whether an s or a pband is the
wider. Discuss the thermal conductivity of the lattice you
have just considered. What is the nearlyfree electron approxi
mation? Why does the presence of a periodic lattice introduce
discontinuities into the energy versus k dependence? Discuss
how these bands are related to the tightbinding bands. What
is a Brillouin zone? Discuss the conditions for a material to be
a metal on the NFE theory. What is meant by the terms 'Fermi
energy', 'Fermi surface', and 'Fermi sea'? What explanation of
the heat capacity of metals is provided by the band theory?
benzene
19
Suppose that an empty band is within an energy kT of the
upper edge of a completely filled band: discuss the depen
dence of the electrical properties on the temperature of this
semiconductor,
2. Treat the onedimensional chain of N atoms as a "Htickel
problem and show that as atoms are added to the chain the
levels gradually form a continuous band, but of finite width.
Proceed by setting up a "secular determinant based on a
resonance integral jS between neighbours, and between non
neighbours. Let there be N atoms in the chain. Show that the
roots of the equation lie at 23eosl/J7T/(rtrT )] , n = 1 , 2 /V,
and plot the energy levels for N = 1, 2, 3, . . . 12, °°.
Further information. A simple account of the structure of
metals and insulators and the way the theory may be used to
account for a wide range of properties is given in Chapters 6
and 7 of Solymar and Walsh (1970) and in Jennings and
Morris, Atoms in contact (OPS 4). Then see Altmann (1970)
for a more quantitative account, but still at a moderately
elementary level, and Quinn (1973), who gives an account of
the TBA and NFE approximations and their more sophisti
cated developments. See also Kitte! (1971), Ziman (1972),
and Dekker (1960). Semiconductors are simply described in
Solymar and Walsh (1970). Amorphous materials are of
immense technological importance, and a summary of this
difficult field has been given by Mott and Davis (1971 (.
benzene. Benzene is the archetype of the "aromatic mole
cules in which the ring of carbon atoms is unusually stable
when compared with other unsaturated systems. The reson
ance theory of chemical structure is the vale nee bond (VB)
attempt to explain this stability, and the derealization picture
is the molecularorbital (MO) attempt.
The classical picture of the structure of benzene is that of
a hexagon of alternating single and double bonds. The energy
of this Kekute structure may be calculated by taking into
account all the electrostatic interactions of the electrons and
nuclei. But the interelectmnic repulsions have the further
effect of pushing the irelectrons into the vacancies between
the double bonds; that is, there is a tendency for the molecule
to turn itself into the other Kekule structure (Fig, B8). If the
transformation is not allowed the molecule has an energy E ; if
FIG. B8. Some of the electrostatic repulsions are relaxed by permitting
resonance between two Kekule' forms.
the transformation is allowed the energy falls below E because
the structure is more relaxed. The "resonance picture of the
stabilization may therefore be interpreted in terms of the
lowering of energy brought about by permitting the molecule
to 'resonate' between the two Kekule structures. This picture
of two resonating Kekule structures loses its edge when it is
realized that these two structures contribute only about
80 per cent of the true total structure (which is a "super
position of resonating structures), the remaining 20 per cent
being due largely to Dewar structures (Fig, B9). The 'reson
ance' picture gets rather muddy at these depths, although it
Conventional representation
Kekule form f 80/ )
Dewar form
C20 % J
FIG. B9. Five resonating forms contributing to the VB structure of
benzene. The true structure, designated by its conventional represen
tation, is a superposition principally of these fundamental structures.
20
benzene
is valid when treated properly, and further details are given
under "resonance.
The "molecularorbital theory gives the impression of
providing a more transparent description of the stability of
the ring, based on the fact that benzene is an even "alternant
molecule which may be described in terms of the orbitals and
energies shown in Fig. B10. Six electrons are to be added to
FIG. B10. Hiickei energy levels and orbitals for benzene. The numbers
are the coefficients of 2p for each atom; the sign of the coefficient is
marked on the orbital. Obtain unmarked coefficients by symmetry.
the del oca I i zed 7rsystem, and, as the first three orbitals are
bonding, all the added electrons contribute to the stability of
the molecule.
The 0"electrons have an important role to play in both the
VB and the MO descriptions because six carbon atoms sp 
"hybridized (with lobes at 120°) form a sixmembered ring
(with angle 120°) without strain. Each of the six carbon atoms
has its full share of its valence electrons, and so not only is
there no energy arising from an imbalance of charge, but there
are also no centres of charge excess or deficiency to provide a
reactive site (see "alternant hydrocarbon). The hexagonal ring
of carbon atoms is a very well poised system.
Questions, 1. Why does the "resonance of two Kekule' struc
tures lend stability to the benzene ring? What structures must
be taken into account in the full vatencebond treatment, and
what proportion do they constitute of the total structure?
Should ionic structures be included too? Draw some. What
features of the hexagonal structure of benzene make the
molecule stable in molecuiarorbita! terms? What features of
this explanation are equally applicable to a valencebond
description?
2. Find combinations of the three occupied benzene molecular
orbitals that may be interpreted as the three localized bonds
of a Kekule structure. (See localized orbitals.) How may the
other structure be obtained from the molecular orbitals? How
may the Dewar structures be obtained?
3. Use the symmetry of the molecule to find the molecular
orbitals in the Huckel scheme, and find their energies. Find
the "resonance energy of the molecule. The experimental
value is 150 kJ mof 1 (how would you determine this
experimentally?).
Further information. See MOM Chapter 9 for the simple
molecuiarorbita! description of benzene. A simple account
of the structure of benzene is given by Co u I son in his The
shape and structure of molecules (OCS 9) and in Chapter 9 of
Coulson (1961). For the valencebond description see also
Pauling (1960), For more mathematical details of the cal
culations see McGlynn, Vanquickenborne, Kinoshtta, and
Carroll (1972), Salem (1966), and Pilar {1968}. For a recent
review of the spectroscopy of benzene see Murrell (1971) and
Hall and Amos (1969). Other entries of interest are "resonance,
birefringence: a synopsis
21
"aromaticity, "Hu'ckel method, and "alternant. Some physical
properties of benzene are collected in Table 1.
birefringence: a synopsis. Birefringence denotes the
presence of different "refractive indexes for the two polar
ization components of a beam of light. Birefringence may be a
natural property of isolated molecules, or of a particular
crystal form, or it may be induced electrically, magnetically,
or mechanically. An important example of natural birefring
ence is optical activity in which the plane of polarization of
light is rotated as it passes through the medium. This is an
example of circular birefringence, because the effect depends
on the different rates of passage of the left and right circu
larly polarized components (remember that the velocity of
light in a medium hc/n). The Faraday effect is the induction
of circular birefringence in a naturally inactive sample by the
imposition of a longitudinal magnetic field. The rotation
induced is proportional to the field strength, and the constant
of proportionality is the Verdet constant; all molecules give a
Faraday rotation. A magnetic field applied transversely induces
a linear birefringence in which one planepolarized component
travels faster than the other: this induces an ellipticity into the
beam, and is referred to as the CottonMouton effect. The
same phenomenon induced by a transverse electric field is
called the Kerr effect Because the "refractive index and the
absorption coefficient are related (by the Kramers Kronig
dispersion relations), and when one depends on the polar
ization so does the other, both the Kerr and the Cotton
Mouton effects may be observed by monitoring the different
absorption coefficients for the polarization components.
Birefringence may be induced by fluid flow if the molecules
are sufficiently anisotropic: the alignment of long thin mole
cules introduces an anisotropy into the optical properties of
the medium and the refractive indexes depend on the orien
tation of the polarization of the beam: this is streaming
birefringence.
Further information. See MQM Chapter 1 1 for a discussion of
the quantummechanical basis of natural optical activity. A
further simple discussion has been given by Kauzmann (1957),
and a thorough review, which also deals with induced birefrin
gence, is in the book by Caldwell and Eyring (1971). See
Fredericq and Houssier (1973) for a simple account of electric
drchroism and electric birefringence, especially their appli
cations. Other reviews containing interesting material are those
of Moscowitz (1962), Tinoco (1962), Mason (1963), and Urry
(1968). For a further discussion of the Faraday and Cotton
Mouton effects, see Buckingham and Stephens (1966). For the
application of induced birefringence in liquid crystals to dis
play devices, see the article by Elliott (1973). For the optical
properties of solids, see the standard work on optics by Born
and Wolf (1970), Landau and Lifshitz (1958a), and Wooster
(1973). The dispersion (the frequency dependence) of optical
activities is a basic research tool for the study of the stereo
chemistry of molecules: for an introduction to optical
rotatory dispersion (ORD), see Crabbe (1965).
blackbody radiation. A black body is one that absorbs all
the radiation incident upon it. A practical example is a con
tainer completely sealed except for a tiny pinhole: this hole
behaves as a black body because all light incident on it from
outside passes through and, once in, cannot escape through
the vanishingly small hole. Inside it experiences an indefinitely
large number of reflections before it is absorbed, and these
reflections have the result that the radiation comes into
thermal equilibrium with the walls. Within the cavity we can
imagine the electromagnetic field as having a distribution of
frequencies characteristic of the temperature of the waits. The
presence of the hole enables a small proportion of this equi
librium radiation to seep out and be detected, and the distri
bution of wavelengths in the blackbody radiation is the same
as the distribution within the equilibrium enclosure because
the pinhole is a negligible perturbation. Blackbody radiation is
the radiation in equilibrium with matter at a particular temper
ature.
The general features of the distribution of frequencies are
familiar from everyday experience: at low temperatures the
hole does indeed look black, but a sensitive detector would
show that a small amount of long wavelength radiation is
present. At higher temperatures the amount of energy
emitted is much greater, and its principal component lies in
the infrared. At still higher temperatures the pinhole glows
dull red, then white, and afterwards blue, and the total
amount of energy radiated increases dramatically. As the
22
black body radiation
temperature is raised further the peak of radiation passes
through the ultraviolet, although the temperatures at which
this occurs are too great to be conveniently accessible, There
fore we have made two observations: that the energy present
in equilibrium increases dramatically as the temperature is
raised, and that the wavelength of the light shifts towards
the blue and beyond. The former observation is summarized
by Stefan's Law, that the total energy emitted at a tem
perature T is equal to al* ; the constant a is Stefan's constant
and has the value 567 X 1 0" 8 W m' 2 K" 4 . The second
400
blue
'WO V s " wo
yellow red
b 00
infrared
5000 K
~~ i — i 1 — n — r~
02 04_ Wl 06
MT/he
FIG. Bit. Radiation density from a hot black body at different
frequencies and temperatures.
observation is summarized by Wien's displacement law, that
the wavelength of maximum intensity is inversely proportional
to the temperature: A T= 29 mm K.
r m ax
The notoriety of blackbody radiation lies in its defeat of
classical mechanics, and in its role in the inception of quantum
theory. The RayleighJeans law was the result of applying the
ideas of classical mechanics: Rayleigh, with Jeans's help,
counted the number of oscillators of a particular wavelength
that could be found in a cavity, and then applied °equipar
tition to associate an energy of k T with each mode of
oscillation; the energy density at the wavelength A was pre
dicted to obey the rule 8rrkT/\ 4 (Fig. B11). This conforms
with none of the observations mentioned above: it predicts
an enormous energy density at short wavelengths (called, by
Ehrenfest, the ultraviolet catastrophe) and an infinite energy
in any enclosure (the integral of the energy over all wave
lengths diverges). There is no maximum in the distribution,
except the absurd one at zero wavelength. According to
classical mechanics, even a glowworm would devastate the
surrounding countryside with its radiation.
Planck's quantum hypothesis was a device that saved the
situation even though in the process it abolished classical
mechanics. From thermodynamic arguments Pianck was
forced to the conclusion that he needed a single distribution
law that yielded the RayleighJeans formula at low frequencies,
and the Wien displacement law at high frequencies; therefore
he found a formula which when approximated at low and
high frequencies gave the two taws (see Box 2). On reflecting
on the significance of his formula he was forced to the con
clusion that it represented a situation in which a radiation
mode of a specified frequency could not possess an arbitrarily
small energy: a mode could be excited only in discrete
amounts of energy, or quanta. Whereas the classical law said
that if there is a mode of frequency v in the container, then
with it should be associated its due portion of energy accord
ing to equipartion, namely kT, the quantum law says that, if
we have that mode, it shall be excited only if It is supplied
with at least its due amount of energy, namely hv. The con
stant h is now referred to as Planck's constant (66256 X
10 34 J s). This means that the very high frequency modes
are not excited at normal temperatures because the thermal
energy present in the walls of the container is insufficient to
blackbody radiation
73
pM
BOX 2: Planck distribution law for blackbody radiation
Energy density {energy per unit volume) in the range dX at
the wavelength X:
dU(K) = p(X)dX
nt \\ (MiA j expthcfSkT)
PW \ X s j ] ^x 9 lficf\kT)\ ■
Energy density in the range dv at the frequency V.
dU(v) = p(v)dv
( sitfilA j exp(ftf/frr) 
\ c 3 ) jlexplfiWttTlf '
RayfeighJeans law (longwavelength, lowfrequency limits
of above):
p(X)~8nk77X 4 \>hclkT
piv) ~ 8m> 2 * 77c 3 v < *T//j.
Stefan's taw (total energy density <* T 4 ):
y = ^ dXo(X) = JTdvp(f) = io/c)T*
O = 56697 X 10" 8 W m" 1 K~" .
Wien '$ displacement law (X_ T = constant) :
*^ max
X T=hclbk = b
max
6= 28978 X 10" 3 m K.
supply them with adequate energy (see quantum). This
damping effect on the highfrequency oscillators quenches the
rise of the RayleighJeans distribution at high frequencies
(short wavelengths), and so it eliminates the ultraviolet
catastrophe. Furthermore, it introduces a maximum into the
energy distribution versus wavelength curve, and this is in
accord with the Wien law at high frequencies (and the Wien
constant for the maximum is found to be equal to hc/bk).
Because of the elimination of the highfrequency excitations
the total energy emitted at a temperature T is finite, and the
Stefanlaw dependence on T 4 is reproduced. Thus we are
forced to accept the propriety of Ptanck's quantum hypoth
esis; and, since the interaction of matter and radiation is such a
fundamental process, it should be no surprise that the ramifi
cations of the hypothesis affected the whole of our appreci
ation of the nature of the world.
Questions. 1 . What is a black body, and how may it be
realized experimentally? Why is the radiation emitted through
a pinhole in an otherwise closed cavity of fundamental
importance? What are the changes in the frequency distri
bution of light emitted by a black body as its temperature is
raised? State Stefan's law and Wien's law. What is the basis of
the classical calculation of blackbody radiation, and why is its
form unacceptable? What was Planck's contribution, and what
is the effect of quantization on the highfrequency oscillators?
Discuss the differences between the excitation of a classical
and a quantum "harmonic oscillator. Calculate the high
frequency form of the Planck distribution law (Box 2) and the
shortwavelength form. From the latter show that the distri
bution passes through a maximum inversely proportional to 7",
and deduce an expression for the Wien constant, evaluating it
numerically. Calculate the energy density radiated by a black
body at wavelengths of 1 cm, 55 nm, and 200 nm when it is
heated to 300 K, 1000 K, and 10 s K, Suppose that the fila
ment of an incandescent lamp is a blackbody radiator (the
approximation is not absurd) and calculate the temperature to
which it must be raised to emit light predominantly in the
visible region of the spectrum.
2. This question concerns the determination of the distri
bution function, and falls into two parts; the first (which we
shall do) inserts the quantum hypothesis into the Boltzmann
distribution to get the mean energy of a mode, and the
second (which we shall not do) counts the number of modes
of a particular frequency. If the energy of an oscillator is
confined to the values nhv, then the probability that the
oscillator has an energy nhv is, according to Boltzmann,
ex.p(—nhv/kT)/Z, where Z is the partition function, or
^n»o exp l — nhvlkT) . The mean energy is the sum of
[(nhv)exp(— nhvikT)IZ] over all the values of n (n — 0, 1,
2, . . .). First evaluate Z by realizing that it can be written as
the sum over x", with x suitably chosen, and so it is a geo
metric progression; and then evaluate the sum over the
numerator by realizing that it can be related to dZ/dT. Hence
find the mean energy of a mode of wavelength X = civ. The
number of modes of radiation in the range dX at the wave
length X is 8"JTdXA 4 ,' hence find the Planck distribution law.
The answer is quoted in Box 2.
7A
Bohr atom
Further information. See MQM Chapter 1 for a derivation of
the Planck distribution and the number of modes in a con
tainer. For the latter also see p. 41 of Heitler (1954), §2,1 of
Power (1964), §1.4 of Bohm (1951), and p. 144 of Lin
(1967). Blackbody radiation is discussed in an historical
perspective by Jammer (1966), and a recent review is that of
Lin (1967). See also Ingram's Radiation and quantum
physics (OPS 2) and §9.13 of Re if (1965) for a useful discussion.
Numerical values of the Planck distribution over a wide range
of temperatures and wavelengths, together with integrated
intensities, will be found in the American Institute of Physics
handbook, Gray (1972), p. 6.198 and in Abramowitz and
Stegun (1965).
Bohr atom. Before the discovery of quantum mechanics,
Bohr applied the principle of quantization to the problem of
the structure of the "hydrogen atom. He asserted that:
(1) an electron remained in a stationary state until it made
a transition;
(2) a transition from a stationary state of energy E . to
another of energy £ was accompanied by the emission or
absorption of radiation with a frequency v determined by the
condition hv = E — E . (this assertion is the Bohr frequency
condition);
(3) the permitted stationary states were to be found by
balancing the nuclear electrostatic attractive force against the
centrifugal effect of the angular momentum of the electron in
its orbit.
The quantum condition was imposed at the last stage, for
Bohr asserted that
(4) the only angular momenta permitted were those whose
magnitude was an integral multiple of h.
The calculation of the energy levels done on the basis of
these postulates led to an expression of the form E = —Rln 2 ,
where R is the ■ Rydberg constant and n is the principal
"quantum number {n = 1 , 2, . . .). This expression is in
virtually exact agreement with experiment, and was cause for
jubilation.
Refinement of this promising model proceeded in three
steps. The first took into account the fact that the orbital
motion occurred about the centre of mass of the system rather
than about the nucleus itself: this merely involved replacing
the mass of the electron in the Bohr formula by the reduced
mass tl = m m l[m +m }, The second step was taken by
' b p e p
Sommerfeld: in the BohrSornmerfeld atom the orbits are
allowed to be elliptical, and the degree of ellipticity is
determined by a further quantum number k, the azimutbal
quantum number: but the energy of the orbits was found to
be independent of their eccentricity. The third improvement
was also made by Sommerfeld: he incorporated relativity into
the model, and found that its effect was to cause a mismatch
of the ends of the elliptical orbits, so that the electron de
scribed an open orbit around the nucleus— a continuously
evolving orbit that resembled a rosette. The inclusion of
relativity caused the energies to depend weakly on k, and
quite remarkable agreement with experiment was obtained
(the numbers obtained are the same as those obtained in the
"Dirac theory of the hydrogen atom).
Although the numbers are almost exact, the model of the
hydrogen atom from which they are obtained is fundamentally
wrong, and we are forced to the view that the agreement with
experiment is an astonishing coincidence: this coincidence
probably stems from the very peculiar properties of the
Coulomb potential, properties that remain even in modern
quantummechanical theories of the atom. The fallacy in the
model was indicated by the later developments of quantum
theory, for there it is discovered that the concept of trajectory
is alien to phenomena on an atomic scale (see "uncertainty
principle). Therefore it is wholly false to attempt to discuss
the dynamics of a system in terms of the trajectories of its
components: the Bohr orbits and the BohrSommerfeld orbits
are macroscopic concepts that have no meaning on the scale of
the hydrogen atom. Furthermore, it is quite clear that the
Bohr model is incomplete in the sense that in its postulates it
virtually asserts the structure of the hydrogen atom and no
justification is given for the stationarity of states and the
quantization of angular momentum; these are provided much
later by the theoretical structure of quantum mechanics.
Nevertheless, Bohr's achievement was considerable, for it
applied to a problem in mechanics a theory that had been con
structed on the basis of the behaviour of radiation, and was
therefore one of the first germs of the view that optical and
mechanical phenomena were essentially identical.
bond
25
Questions, I. State the postulates of the Bohr theory of the
hydrogen atom. Which of them conflicted with the require
ments of classical mechanics? What is the form of the ex
pression for the energy that is obtained on this model?
Deduce an expression for the frequency of the transitions of
the hydrogen atom: do these conform with the known
"hydrogen atom spectrum? How was the original mode! re
fined? What is the significance of the quantum number k? The
elliptical orbitals with k = (which are straight lines swinging
through the nucleus) were rejected as implausible: can you see
the connexion of, these rejected orbitals with the sorbitals of
the modern theory of the atom? The lower energy state of the
Bohr theory requires the presence of an angular momentum to
repel the electron from the nucleus: does the quantum
mechanical theory lead to the same conclusion? Why is the
Bohr theory untenable? Discuss the role of the uncertainty
principle in the Bohr theory. Why was the theory so
important?
2. Deduce the energy of the hydrogenatom stationary states
on the basis of the Bohr theory. Set up an expression for the
potential energy of the electron, and then relate the
centrifugal force to the angular momentum. Balance the two,
and then replace the angular momentum by nli. Compare your
result to the accepted expression for the °R yd berg constant.
Further information. See § 1.2 of Herzberg (1944} and Chapter
2 of Pauling and Wilson (1935( for information „bout calcu
lations on the hydrogen atom. For a view of the hydrogen
atom in an historical perspective, see Jammer (1966).
bond. The nature of the chemical bond— the reason why
atoms stick together and form molecules of a definite shape
and energy— is one of the central topics and successes of the
application of quantum theory to chemistry. Elementary
chemistry distinguishes three kinds of bond between atoms:
the ionic (where electrons are transferred between atoms and
the bond is the electrostatic interaction between ions), the
covaient (where electrons are donated by both partners, and
shared more or less equally), and the dative (where one
partner donates both electrons, which then are shared).
Modern quantum chemistry shows how these three types may
be considered to be special cases of a general form of bond.
Elementary chemistry also ascribes the tendency to bond to
the tendency of atoms to 'complete their octet' or to achieve
an 'inert gas configuration'. Modern quantum chemistry
interprets these rules of thumb in terms of the quantum
mechanical properties of electrons and nuclei.
Atoms group together and form molecules if by so doing
the system attains a lower, more favourable energy; therefore
we must seek the reason why energy is reduced when a
molecule is formed. Generally the different stereochemical
configurations of atoms differ considerably in energy from
each other, and so the shape of the molecule is normally
well defined, and corresponds to the stereochemistry that
enables the system to attain its lowest energy. Conversely, a
molecule falls apart into its components, or groups of com
ponents, if enough energy is injected into the structure (by a
collision with another molecule or wall, or by the absorption
of light), so that its total energy exceeds that of its separated
components. Energy determines everything: to understand
the shape and stability of a molecule we must study its energy.
Why energy is important is of course a much deeper problem.
In order to assess the contributions that lead to a lowering
of energy we should remember that the energy consists of
two parts, kinetic energy and potential energy, A careful
analysis of the contributions these make is very difficult, and
quite often people ignore the contribution of kinetic energy
by presuming that it can look after itself, or that the dominant
contribution to binding energy is the lowering of the potential
energy that occurs when electrons and nuclei are brought
close together. Ignoring the kinetic energy is a dangerous
game; and makes the description of chemical bonding look
simpler than it really is. Nevertheless, we shall play the game
because the situation has been fully analysed in only one
place, to which we return later.
Given a kit consisting of two protons and one electron,
where should the electron be put in an attempt to form a
stable molecule (of Hj)? The conventional argument is that
simple electrostatics suggests that the electron should be
placed in the intemuclear region; then the internuclear
Coulombtc repulsion will be overcome by the attraction
between the electron and each nucleus, and the H species
will achieve stability on this account. Naturally the electron
does not congregate solely on the midpoint itself, and the
26
bond
structure should be envisaged as a distribution of charge
around the two nuclei, with a significant accumulation in the
internuclear region. The addition of a second electron, to
form H 2 , will lead to a stronger bond if it too enters the same
region of space, so that the nuclei can now stick to a double
helping of opposite charge.
On the basis of the preceding analysis the structure of the
hydrogen molecule may be envisaged as two nuclei surrounded
by a charge cloud of two electrons, with an accumulation of
charge density in the internuclear region, the bonding region.
The characteristic bond length of H 2 is the point at which an
equilibrium: is reached between the repulsive interaction of the
nuclei, which increases as the bond shortens, and the attractive
interaction with the internuclear electrons; at very short bond
lengths the electrons cannot accumulate in the bonding region
and so the repulsion dominates. If a third electron is added to
an H 2 molecule it attempts to cluster close to the nuclei, but
cannot penetrate the bonding region because of the presence
of the original pair of electrons (it is excluded by the 'Pauli
principle). It therefore congregates as a fuzzy accumulation
outside the internuclear region. The force this electron
exerts on the nuclei is disruptive, and so its presence tends
to lengthen and weaken the bond. A fourth electron succeeds
in breaking the bond.
On this description of the covalent bond ', where the bond
ing electrons are provided equally by the two atoms, it is clear
that two electrons give the strongest bond. Forming strong
bonds is energetically favourable, and therefore atoms tend to
form as many as they can without drawing on the inner, tightly
bound electrons. This situation is what should be held in mind
when one makes the remark that 'atoms share electrons in
order to complete their octets'. Notice also that the two
electrons have to occupy the same region of space to be
effective in bonding, and in order to do so their spins must be
opposed: this is a consequence of the Pauli exclusion prin
ciple. This feature underlies the importance of the electron
"pair in chemical bonding. Electrons do not seek to pair for
some transcendental reason, nor because they lose repulsive
energy by pairing— in fact it requires energy to push two
electrons into the same orbital: they pair in order to attain a
distribution that leads to the lowest energy for the system, and
at the bottom of a stack of distributions often lies a molecule.
This description of the role of an electron pair is seen very
clearly in both the "molecularorbital and °valencebond
theories of molecular structure.
In a heteronuclear bond (a bond between two different
atoms) the situation is analogous, but is modified by the
possibility that the energy will decrease if the bonding pair of
electrons accumulates closer to one atom than the other: they
congregate more in the vicinity of the more "electronegative
atom. This situation may be envisaged in terms of prising off
one of the valence electrons of the less electronegative atom
(the atom with the smaller "ionization potential) and shifting it
towards the atom with the larger 'electron affinity. This
process leads to a polar bond, and in one language (the valence
bond) it would be possible to say that the pure covalent bond
is contaminated by ionic components. (Alternatively we
might say that the molecular °wavefunction is a "superposition
of covalent and ionic wavefunctions.) Atoms with the greatest
electronegativity tend to be those that differ from a closed
shell configuration by only one or two electrons; and so once
again we can understand that the tendency to form an octet is
a manifestation of a search for the lowest energy distribution
of the electrons.
When the electronegativities of the two atoms of a
heteronuclear bond are very different, such as when an atom
of low ionization potential (lefthand side of the periodic
table) Is next to an atom with a high electron affinity (right
hand side of the table) the stability gained by transferring a
whole electron from one atom to the other may be very large,
and the juxtaposition of the two atoms leads to the flow of an
electron from one to the other, so that side by side there is a
positive ion and a negative ion: these stay stuck together
simply by a Coulombic interaction between the charges. This
extreme case of a polar bond is the ionic bond. It is important
to note that all ionization potentials are greater than all
"electron affinities (for atoms), and therefore the Coulombic
attraction between the ions provides the energy for the forma
tion of ionic bonds.
Apart from the polarity of bonds the most significant dif
ference between an ionic and a covalent bond is the directional
properties of the latter in contrast to the lack of directional
properties of the former. This arises because the Coulombic
interaction between two charges is isotropic (the same in all
bond
27
directions), so the structures that can be formed, which are
often extensive aggregates, are governed largely by the steric
problem of packing together ions of various sizes: the ionic
bond gives rise to rigid and extensive crystal lattices with
characteristic packing patterns. In the pure covalent bond the
interaction is by no means isotropic because its strength
depends on the ability of an atom to provide electrons in the
region between itself and its bonding neighbour. For a
diatomic molecule there is no problem in principle, but as
soon as three atoms are considered one encounters the reason
why eovalently bound structures have a geometry determined
by the electronic structure of the bonding atoms rather than
the geometrical problem of packing them together. Taking
oxygen as an example we can understand that the molecule
HO can be formed by the hydrogen and oxygen atoms each
donating an electron to form a polar bond, and then a
second hydrogen atom froms a second bond to yield HOH;
the lowest energy configuration of this molecule occurs when
the second bond is at 1 045 ° to the first because the oxygen
atom can form the strongest bonds at that angle (see •hybrid
ization). At this point the oxygen octet is complete, and the
addition of another hydrogen atom leads to an unstable H 3
molecule {that is H3O possesses more energy than separate
H 2 + H), All that water can do is form a dative bond, where
it supplies both electrons, and this it does to form H 3 0* (in
which the oxygen "lonepair electrons donate towards a bare
proton) oraoxrocomplexes with ions. The water molecule is
therefore a welldefined, discrete entity. The valence of an
atom, the number of bonds that it may form, is moderately
well defined for most covalent compounds, and the stereo
chemistry is determined by the ability of the atom to provide
electrons to attain this valence. This is very clearly brought out
in the "molecularorbital theory of molecular structure and the
theory of "hybridization.
All the preceding discussion is based on the conventional
view in the fourth paragraph; that, it seems reasonably
certain, is a sweet seduction. There are few cases where the
molecular structure has been studied in sufficient detail to
enable the true source of bonding to be analysed critically, but
with Hi it has been possible to draw disconcerting conclusions.
These conclusions run counter to most of the simple accounts
of the chemical bond and, as far as I know, counter to any
thing to be found in textbooks. In Hj the source of bonding
appears to be a subtle interplay between the kinetic and
potential energies of the electrons. As H*and H are brought
together the accurate wavef unction shows that electrons are
indeed shifted into the internuclear region, but that this leads
to an increase in their potential energy, in contrast to the
supposition that their potential energy would decrease if they
could be shared by both nuclei. On reflection, of course, we
should realize that an increase is more plausible than a de
crease, because the lowest potential energy arises when the
electrons are as close as they can get to one or other of the
nuclei,
Where then does the bonding energy come from? First, we
should note that the situation is slightly relaxed by virtue of
the greater domain of freedom open to the electron when two
protons are present, and consequently its kinetic energy due
to its motion parallel to the bond falls; but only at first, for
more complication is to come. A larger decrease in energy
(increase in bonding energy) comes from a contraction which
occurs in the atomic orbitals on each nucleus: this contraction
enables the electron to approach the nuclei more closely, and
so its potential energy falls: this contribution is very large and
is the dominant change in the potential energy. On contraction
there is a price to pay, because as the electron is confined to a
smaller domain its "kinetic energy rises, and this almost can
cels the decrease in potential energy; but not quite, and the
earlier decrease in kinetic energy along the axis helps to
counterbalance the change. Finally, the form of the wave
function shows that there is also a shift of electron density
from regions outside the nuclei into the bonding region as
the atomic orbitals are polarized; this reduces the potential
energy and increases slightly the kinetic energy. The net
effect is a large decrease in the potential energy, which is
dominated by the orbital contraction, and a large increase in
the kinetic energy, also dominated by the orbital contraction,
which does not quite succeed in winning. The overall effect is
that H 2 has a lower energy than H + +H, and so is a stable
species.
We stress that this complicated story has been elucidated on
the basis of a careful study. of the H2 wavefunction, and might
need to be modified for more complex species. But it is an
excellent example of the power of myth in chemistry, and
28
bond order
shows the importance of detailed and accurate calculations in
discovering the true nature of the chemical bond.
Questions. What determines whether atoms will stick
together and form a molecule? What thermodynamic quantity
is the measure of the strength of a chemical bond? What
happens when a large amount of energy is transferred to a
molecule? How may the energy be transferred? What feature
of the distribution of the electron in the hydrogen molecule
accounts for its stability on a simple model? What contribution
to the total energy does this description ignore? Why is the
hydrogen molecule more stable than the hydrogen molecule
ion H^, but H^less stable than both? Can this argument be
extended to the explanation of why two helium atoms do not
form a stable molecule? Indeed, does it apply to the stability
of bonds between all closedshell species, for example the rare
gases? What change in the distribution of the electrons occurs
when a homonuclear bond is replaced by a heteronuclear bond?
Is there an additional contribution to the binding energy?
What determines the extent of polarization of the electrons in
the tend? When dees an almost [jure ionic bond occur? What
is the source of the stability of an ionic crystal: why does not
a crystal of common salt blow apart into a gas of sodium and
chlorine atoms? Why does the crystal have a definite structure?
Why does covalency lead to discrete molecules, and ionic
bonding to extended arrays of atoms? What determines the
shape of ionic and covaient species? What determines the
valency of atoms in covalent and ionic compounds? What
other type of bonding leads to an extended array of atoms
with a structure determined largely by packing consider
ations? Discuss the likely true cause of bonding in H 3 .
Further information. See MQM Chapter 10 for a resume of
bonding theory and more details of its quantum theory. For a
simple account of the structure of molecules see Coulson's
The shape and structure of molecules {OCS9) and Coulson
(1961). See also Murrell, Kettle, and Tedder (1965), and for
an original classic, well worth reading for the way it teaches one
to think about the application of quantum theory to real
chemical problems, see Pauling (1960). A close analysis of the
nature of the reduction of energy when a bond is formed has
been given by Ruedenberg (1962), and this is extended with a
careful discussion of the structure of H 2 by Feinberg,
Ruedenberg, and Mehler (1970). See in particular the analysis
on p. 54 of this reference. For more information about the way
that the concepts mentioned here are developed in quantum
mechanics, and therefore made amenable to quantitative cal
culation, see the entries on "molecularorbital theory,
°valencebond theory, "antibonding, electronegativity,
'hybridization. One type of bonding ignored in the discussion
was that responsible for the structure of metals: see "band
theory of metals for a short account.
bond order. The bond order is a measure of the single,
double, or triplebond character of a bond. In °valencebond
theory it is determined by calculating the proportion of single,
double, and triple bonds in the contributing structures. In
molecularorbital theory, which now provides the more
common definition, it is defined in a slightly more subtle
manner. The basis of the definition (which is enough for a
qualitative understanding) relies on the cancellation of the
effects of occupied bonding and "antibonding orbitals. Thus
in Hj, where one Obond is fully occupied, the Obond order
is unity; in Hei, where the Obond and its antibonding
counterpart are both fully occupied, the bond order is zero.
In ethene a Obond and a TTbond are fully occupied, and so
the overall bond order is 2 (a 'double bond'/, and in oxygen
one TTbond is cancelled by a TTantibond {see "molecular
orbital theory for details) and the rump, one O and one
TTbond, gives an order 2. This idea may now be made more
sophisticated in order to accommodate fractional bond orders.
In the °LCAO description of molecular orbitals a bond is
formed when two atomic orbitals on neighbouring atoms
overlap and interfere constructively. If the coefficient of the
orbital on atom A isc« and on atom B is Cg, then the con
tribution to the bonding will be proportional to the product
c.Cn. The order of the bond between A and B is then defined
as the sum of c.Cn overall the occupied orbitals. When c^ and
c R are simultaneously large for a particular orbital a large con
tribution to the total order results (but it cannot exceed unity);
when c. and eg have opposite signs, corresponding to an
"antibonding character between A and B, the product is
negative— it subtracts from the overall result and so reduces
the bond order. As an example, each C— C bond in "benzene
has a contribution of = from the TTorbitals, and 1 from the
BornOppenheimer approximation
29
oorbitals, and so each bond is of order lE. The C— C bond in
ethane is of order 1, in ethene 2, and in ethyne 3, in accord
with a norma! chemical appreciation of the order of the bonds.
One principal application of the quantitative definition of
bond order is to the estimation of bond lengths, especially of
C— C bonds, An empirical relation has been found between the
length R and order 6qq of C— C bonds, and the length satisfies
the rough rule /? cc = T665 01398 (1 + b QC ), with ff cc
in angstroms (see Table 2), The correlation should take into
account the different hybridizations of the atoms involved in
the bond; the effect of so doing is that for a given bond order
the bond length decreases by 004 A on going from sp 3 to sp 2
hybrids, and then by a further 004 A on going to sphybrids.
A further application of bond order is to the definition of
"free valence.
Questions. What is the valencebond definition of bond order?
Calculate the bond order in benzene on this basis, first on
the assumption that only Kekule structures contribute, and
then on the assumption that 20 per cent of the total structure
is Dewarlike, What is the molecularorbital definition of
bond order? What contribution does an antibonding orbital
make? What is the bond order in H 2 and He 2 ? What is the
bond order in Hj? On the basis of the molecularorbital
coefficients given in Fig. B10 on p. 20 calculate the bond
order for benzene, benzene", and benzene*. Estimate the
bond length of naphthalene in which the TTbond orders are as
follows: 0725 (for 12), 0603 (for 23), 0554 {for 19), and
0518 {for 910).
Further information. See Coulson's The shape and structure of
molecules {OCS 9) and Coulson (1961). A helpful and lengthy
discussion is provided in §6.7 of Streitweiser (1961 ). See also
Coulson (1959) in the Kekule symposium and Pilar (1968).
BornOppenheimer approximation. The Born
Oppenheimer approximation assumes that the electronic dis
tribution in a molecule can be evaluated in a static nuclear
framework. The assumption is based on the great differences
of mass of the electrons and the nuclei: it is assumed that if
the nuclei move the electrons can adjust their distribution
instantaneously to take into account the new potential, and
that the nuclei are insensitive to the rapid fluctuations of the
electrons in their orbitals.
The practical effect of the approximation is that it is poss
ible to simplify both the discussion and the calculation of
molecular electronic structures. Instead of having to treat all
the particles in the molecule on an equal footing, it is possible,
according to the approximation, to set the nuclei into a frozen
conformation, and then to calculate the electronic energy and
distribution corresponding to it. The nuclei can then be moved
to a new conformation, and the electronic calculation repeated.
In this way it is possible in principle to calculate the energy for
all possible arrangements of the nuclei, and then to find the
one corresponding to the lowest energy— the stable confor
mation of the molecule.
The BornOppenheimer approximation makes the molecular
potential energy curve a meaningful quantity: as the nuclear
conformation is changed the molecular energy also changes,
and the dependence of the energy on the conformation is the
molecular potential energy curve. For a diatomic molecule
the curve is a plot of energy against bond length, and for a
polyatomic molecule the curve is a complicated potential 
energy surface. Such a curve corresponds to a potential
energy because if the molecule is released from a non
equilibrium conformation it wilt spring back into equilibrium
(or at least vibrate around the equilibrium point), and so the
rise of energy with changing conformation corresponds to the
acquisition by the molecule of a potential energy. It should be
clear that this description relies upon the validity of the Born
Oppenheimer approximation, for only then are we able to talk
about the molecular energy as a function of the parameter
determining the conformation of the molecule (bond length
for a diatomic). If the BornOppenheimer approximation were
to fail (tf we were dealing with light or rapidly moving nuclei)
the notion of a potentialenergy surface would fail, and so too
would the idea of bond length and bond angle. In practice, the
approximation fails slightly, and small spectroscopic conse
quences are observed.
Questions. State the BornOppenheimer approximation. Upon
what is it based? When might the approximation fail? What
simplification does it introduce? Discuss the concept of a
molecular potentialenergy curve for a molecule. Calculate
30
boson
the relative velocities of an electron and a proton each with a
kinetic energy of 100 kJ mof '.
Further information. See MQM Chapter 9 for a brief dis
cussion; for an account of the approximation with some
mathematics see p. 252 of Slater (1963). For spectral conse
quences of the failure of the approximation see King (1964).
The original paper is Born and Oppenheimer (1927).
boson. A boson Is a particle possessing an intrinsic °spin
"angular momentum characterized by an integral spin
quantum number, including zero. Examples include the
deuteron 2 H {/ = 1 ), the 4 He nucleus, or ocparticle, the 4 He
atom (/ = 0), and the "photon [/ = 1). Bosons are not re
stricted by the "Pauli exclusion principle (in contrast to
•fermions), and any number may occupy a single quantum
state. They do obey the "Pauli principle itself, which demands
that a 'wavefunction be symmetrical under the interchange
of any pair of identical bosons. Because many bosons may
occupy a single state, at low temperatures peculiar properties
arise; these include superfluidity and superconductivity (where
pairs of electrons, fermions, behave like bosons). The operation
of "lasers depends on the "photon being a boson, for an
intense monochromatic beam of light consists of a large
number of photons in the same state.
Further information. See "spin and the "Pauli principle for a
further discussion. The table of nuclear properties on p. 277
reveals at a glance which nuclei are bosons and which fermions.
The way bosons occupy states is taken into account by the
BoseEinstein statistics which are discussed in Gasser and
Richards' Entropy and energy levels (OCS 19), and in
Chapter 6 of Davidson (1962), §9.6 of Reif (1965}, and
Chapter 22 of Hill (1960). For a discussion of the fundamen
tal role of the distinction between fermions and bosons see
the article by Peierls in Salem and Wigner (1972) and also see
Pauli (1940). For accounts of superfluidity see Chapter 15 of
Rice (1967), and for superconductivity see Chapter 1 1 of
Kittel (1971 ) and Roselnnes and Rhoderick (1969). The
question of whether the fermion 3 He can show superfluid
characteristics {if two stick together) is discussed by Osheroff,
Gully, Richardson, and Lee (1972). Peierls (loc. cit.) also
discusses the evidence for all particles being either bosons or
fermions.
bracket notation; bra and ket. The bracket notation,
which by virtue of the division {bra Id ket) gives its name to
the entities known as bras and kets, was introduced by Dirac.
The state of a system whose wavefunction is $ (r) is rep
resented by the ket \n), and the conjugate $/*{r) by the bra
</j. The integral /dr^{r)t// {r) is implied by the symbol
(m\n), and the integral fdrip* {r}£lij/ {r) by the bracket
(m\Q,ln),il being some "operator. This elegant notation shows
very clearly the connexion of wave mechanics with "matrix
mechanics, and enables the whole of quantum theory to be
put on deep structural foundations, for the kets may be
interpreted as vectors in a special kind of space (Hilbert space).
Further information. See MQM Chapter 4 for the use of the
notation. An account has been given in §6 of the book by
Dirac (1958) and discussed further by Jauch in Chapter 9 of
Salem and Wigner (1972). The subject is a component of the
structure known as transformation theory or representation
theory: see Chapter 4 of Davydov (1965), Roman (1965),
von Neumann (1955), Katz (1965), Kaempffer (1965), and
Jauch (1968).
branch. The "rotational transitions that occur when a mole
cule makes a "vibrational transition give rise to a structure in
the spectrum which can be grouped into branches: when the
rotational state of the molecule changes from J to J— 1 the
tines constitute the Pbranch; when J is unchanged the lines
constitute the Qbranch; and when J changes from J to J+\
the lines constitute the Rbranch. In "Raman spectra the
vibrational transitions may be accompanied by changes of
± 2 in the rotational quantum number: the resulting lines
form the Obrancb and the Sbranch (for J — * J— 2 and
J — >J+2 respectively). The Qbranch of vibrationrotation
spectra is absent when the molecule lacks a component of
angular momentum about its symmetry axis: thus almost all
diatomic molecules show no Qbranch (the exceptions are
those with a component of orbital electronic angular momen
tum about the internuclear axis, such as NO). The appearance
branch
31
IT
J
4
1
1
!
i
1 !
II.
i
1 E 1
i i I
4
!
1
i ' 1
i i i
i i i
i i
i
3
i i !
i i !
i i !
i i i
i i i
2
i 
i i
1 i
1
1 1
R —
n
1
i
i i
■ i i
i ]
Q
head
FIG. B12. Formation of P, Q, R, branches; note the head on the
Rbranch which arises when the rotational constant in the upper level is
smaller than in the lower.
and source of the P, Q, and Rbranches are illustrated in
Fig. B12, and the energies of the transitions are given in Box 3.
When the upper and lower vibrational states have different
moments of inertia (which is especially likely when the upper
vibrational state belongs to an electronically excited molecule).
the spacing of the branch lines changes with the value of J. In
a diatomic molecule it is common for the moment of inertia
of the upper state to exceed that of the lower, and in this case
the lines of the Rbranch converge as J increases, and may even
pass through a head at high frequency. The reason for this
behaviour can be traced to the [B 1 — B")J 2 term in Box 3,
which outweighs (36' — S'V at high J. An Rbranch show
ing this behaviour is said to degrade to the red. When the
moment of inertia is smaller in the upper state the Pbranch
shows the head and degrades to the violet.
The method of combination differences is used to extract
the "rotational constants of the upper and lower states.
Choosing the appropriate pairs of lines in the R and P
branches (see Box 3} and plotting their energy differences
against J" gives B' or B" from the slope of the line.
Questions. What is meant by a branch in a vibrational
spectrum? What is the classification of the branches? To what
rotational transitions do they correspond? When does a mole
cule show a Qbranch? Why should the Qbranch consist of a
Notation:
BOX 3: Branches
J
J"
B'
B"
rotational quantum number of upper states
rotational quantum number of lower states
rotational constant of upper state
rotational constant of lower state
hv energy of vibrational transition.
Rbranch (high energy; AJ = +1 ;J =J + 1 1
A£ = hv + 2B' + (38'  B")J" + {$'  B")j" 2 = hP + RU")
Qbranch (&/ = 0;/ =/)
t£ = h»+\B'~ B")J' + (ff'  B")/ 2 «*f>V+ Qtf)
Pbranch (low energy; £J — — 1 ; J' = J — 1 )
££=h»[B' + B")J" + (6'  B")f 2 =hv + P{J")
If B' < B" the Rbranch may form a head; if B' > B" the
Pbranch may form the head.
Combination differences
mj")py") = 4B'v" + h
rv"})pu"+\) = 4b"u" + h .
32
Brillouin's theorem
set of very closely spaced lines? On the basis that the energy
levels of a rotating diatomic molecule are given by the
expression E{v, J) = (c+ j)ht<J + B v J[J 4 1), comment on
why the rotational constant B should be labelled with the
vibrational quantum number V, and deduce expressions for
the energy of the transitions of the P, Q, and Rbranches of a
rotating molecule. What spectral information can be obtained
by a study of the branch structure? Now suppose that B
depends on the electronic state of the molecule. Deduce the
expressions given In Box 3 for a transition involving electronic,
vibrational, and rotational excitation, and find the value of J
for which a head may be formed (assume first /' > /" and then
/ < / ' , / being the moment of inertia).
Further information. See MQM Chapter 1 for a discussion of
rotational structure. For an account of the rotationvibration
spectra of molecules, examples of branch structure, and an
account of the information that they can provide, see Chapter
4 of Wheatley (1968), Barrow (1962), Whiff en (1972), King
(1964), and Herzberg (1950).
Brillouin's theorem. Singly excited states of closed she II
molecules do not mix directly with the ground state. (Like
most statements, this can be made to sound more complicated;
for those who enjoy such sounds we may state the theorem as
follows: if a matrix element of the electronelectron Coulomb
interaction is calculated between a closedshell configuration
of a molecule and a configuration differing by the excitation
of a single electron, then that matrix element is zero.)
The delight of this theorem will be found in the simplifi
cation of configuration interaction improvements to self
consistent field calculations, because it implies that the singly
excited configurations cannot contaminate closedshell ground
states by mixing directly into them. But do not interpret that
as meaning that there is no mixing at all with singly excited
configurations, because these may still mix by an indirect
process involving interaction with an intermediate state. In
some cases, indeed, it is found that the indirect route is so
effective that the singly excited configurations are very
important.
It should be appreciated that the existence of Brillouin's
theorem implies the stability of the ground state as calculated
by selfconsistent methods; if ii were false then the ground
state could be strongly perturbed by closelying singly excited
configurations; but as it is true, direct mixing occurs with only
relatively distant multiply excited levels, and only indirectly
with the singly excited levels.
Further information. See §63 of Richards and Horsley (1970)
for a simple introduction and proof of the theorem, and
Slater (1963) p. 141 and Appendix 4 for a slightly longer dis
cussion and proof. See Brillouin (1933).
c
character. In chemical applications of "group theory a
"symmetry operation is generally represented by a "matrix.
The character \(R) of the operation R is the sum of the
diagonal elements {trace or spur) of the matrix. (All these
names are interrelated: Spur is the German for spoor, hence
trace and character.) Those who know the basic features of
group theory may wish to be reminded of the following facts:
(1 ) symmetry operations in the same class possess the same
character in an irreducible representation of the group;
(2) the characters of different irreducible representations of
a group are orthogonal (Note 2 in Box 4);
(3) the character of an operation is invariant under a
similarity transformation;
{4) the character of a reducible representation of a group is
equal to the sum of the characters of the irreducible represen
tations into which it is decomposable {Note 3 in Box 4);
(5) the characters may be combined into the form of a
projection operator which when applied to an arbitrary
function projects out a component that is a basis for an
irreducible representation of the group {Note 4 in Box 4);
(6) the character of the identity operation in a particular
irreducible representation is equal to the dimension of that
representation; the dimension of the representation is equal to
the degeneracy of the basis of the representation.
These properties are summarized in mathematical terms in
Box 4; further information is given in Further information,
and a few useful character tables are listed in Table 3 on
p. 266267.
Further information. A sketch of the content of "group
theory will be found under that heading; details of its
content, method, and application will be found in MQM
Chapter 6, and examples of the way it is applied to problems
of molecular structure and properties in Chapters 9, 10, and 11.
A list of character tables for alt the common point groups, and
some of the uncommon ones, with some simple notes on how
to apply them, has been prepared by Atkins, Child, and Phillips
(1970). For a thorough analysis see Cotton (1963), Bishop
(1973), Tinkham (1964), and Bradley andCracknell (1972),
especially for solids. Books introducing the idea of group
theory are listed under that heading.
BOX 4: Character
1 . Definition Y (/) (R) s tr D (/) (fl) = 2 D U) (fl)
ll MM
X 1 ' 1 W is the character of the operation R in the rep
resentation T , in which the operation R is represented by
the matrix d" 1 !/?).
2. Orthogonality X^ n iR)*^ i) {R) = ftS.,
h is the order (the number of elements) of the group.
3. Decomposition of representation T
4. Projection operator for basis f*'* of F 1 " from a general
function f
pW/» *<« p ""> = (yMSx""' l*)*fl
£. is the dimension of I" 1 *''.
5. Selection rules. The integral JdTf i ' ) *Sl l ^f <k) disappears
unless reoccurs in the decomposition of V "' X P .
33
34
charge density
charge density. There are two sorts of charge density: one is
the density of charge at a particular point In an atom or mole
cule, and the other is the charge that may be associated with a
particular atom in a molecule. The former may be determined
if the "wavefunction is known, for the probability of finding
an electron in a small volume element dr surrounding the
point r is simply li/>(r)PdT. Since an electron carries the charge
— e it follows that the amount of charge in this region is
—e\\p{r)\ 2 dT. The charge density (charge per unit volume) at
the point r is therefore —el i/'fr)! 2 . This charge density may be
used for a variety of purposes, for example in the evaluation
of the "Coulomb integral or in the calculation of Xray
scattering properties of atoms.
The other definition lays a much coarser grid on the
molecule: it does not seek to know all the intimate details of
the distribution of the electron at each point of space. The
charge density in a molecule generally means the amount of
charge (or the density of electrons less the number of nuclear
charges) on each atom in the molecule. Thus in a homonuctear
diatomic there is zero charge density on each atom because
the electrons are equally divided between the atoms, and the
nuclear charge exactly cancels the electronic charge. In a polar
molecule electrons may accumulate closer to the more
'electronegative atom; then the charge density is not uniform.
Charge density is often calculated from a wavefunction that
has been written as a "linear combination of atomic orbitals. If
the amplitude of an orbital in a filled molecular orbital is Ca, so
that the proportion of that atomic orbital in the molecular
orbital is \c^\ 2 , then that orbital contributes elc A I 2 to the
charge density on atom A if it is occupied by a single electron.
The total charge density on atom A is calculated by summing
all such terms for the occupied orbitals on the atom. The
charge density may be used to calculate the "dipote moment
of the molecule, and in discussions of its reactivity.
The analysis of the distribution of an electron in terms of
its population of atomic orbitals is known as population
analysis.
Questions. What are the two definitions of charge density?
What is meant by population analysis? How may the charge
density at an atom be calculated if the molecular structure is
known? The wavefunction for the Isorbital of atomic
"hydrogen is i/' ls ('")= (l/JBo^expl— rla ): what is the charge
density at a point r? What is the total charge density of the
atom? The wavefunction for the hydrogen molecule ion is
(1//2) [^ ]s ('" al ) + ^ ls ( r bl )] . where r t is the distance of
electron 1 from nucleus a and r its distance from nucleus b
b]
(note that the molecular orbital is not accurately normalized:
does it matter?): what is the charge density on each atom?
What is the charge density on each atom in the "benzene
molecule?
Further information. See "alternant hydrocarbon for a
further point concerning the CouisonRushbrooke theorem
and charge density; see also "dipole moment for the way the
concept is applied. "Bond order is a related concept of
population analysis, A discussion of charge density will be
found in Coulson (1961); Pilar (1968), McGlynn,
Vanquickenborne, Kinoshita, and Carroll (1972). Tables of
molecularorbital parameters have been prepared by
Coulson and Streitweiser (1965). See also Streitweiser (1961),
Salem (1966), Pople and Beveridge (1970), and Murrell and
Harget (1972).
Closure approximation. The expressions that appear when
in order to calculate a molecular property one indulges in
secondorder "perturbation theory normally contain sums over
all the excited states of the system, and to incorporate them
properly it is necessary to know the energy of every excited
state and the wavefunction for each. Normally this is an imposs
ible task because neither the excitedstate energies nor the
wavefunctions are known at all accurately. When an exact
answer is not required, for example, when an order of
magnitude of a physical quantity is sufficient or when it is
desired to see in a general way how the property depends on
various aspects of the molecule, it is possible to make the
closure approximation. This consists of pretending that all the
excited states that mix into the ground state have the same
energy. The propriety of this we shall ignore for the moment;
but its advantage is that expressions of the form Z/Wof,/V n o/
(£ —En ), which reference to Box 1 6 on p. 1 72 shows to be
central to perturbation theory, are replaced by
(1/A)i M 0n N no where A is the mean excitation energy. This
expression may be developed further by recognizing that it
chemical shift
35
has the form of a "matrix product {but see Questions); and so
the secondorder perturbation expression reduces to
(M/V)oo/A. In order to evaluate this one needs to calculate
only the groundstate 'expectation value of the product MN,
and no knowledge at all of the nature or the energies of the
excited states is required. The step from the sum over /Won^no
to {MWqo is the 'closure' that gives the approximation its
name.
But what price have we paid? The error in the method is
the absurdity that all the excited states have the same energy.
When they all lie close together (as in some sense is true in the
"hydrogen atom, see Fig. G3 on p. 86) the approximation is
not ludicrous. In other cases the method can be justified
weakly by saying that the value of A is to be chosen as a
parameter which relates the true sum to the quantity we can
evaluate (MN)^; the parameter is then varied in a plausible
way to simulate the effect of modifications to the molecular
structure.
Questions. What is the source of the difficulty of using
perturbation theory to calculate molecular properties, and
how does the closure approximation circumvent it? When is
the closure approximation plausible? The sum over the
excited states in the perturbation expression is not really a
true matrix sum because, according to Box 16 on p. 172, the
ground state has to be omitted; show that the closure
approximation really leads to an expression with (/WA/)oo ~~
MqqNm in the numerator, and so only the groundstate
properties are required oven with this correction. When M and
N are identical interpret the numerator in terms of fluctuations
in the property M (interpret the numerator as a meansquare
property). Discuss the closure approximation in terms of its
application to the calculation of electric "polarizabilites, where
more information will be found. Show, in particular, that the
polarizability of a molecule should increase with its size.
Further information. See MQM Chapter 1 for a discussion
and Chapter 11 for applications of the closure approximation
to the calculation and discussion of the electric and magnetic
properties of molecules. Further applications are outlined by
Davies (1967), who underlines, on his p. 47, the remark made
by McLachlan (1960) about possible limitations on the appli
cability of the closure approximation.
Chemical shift. In a "nuclear magnetic resonance (n.m.r.)
experiment an external magnetic field is adjusted until the
energy required to invert a nucleus (to realign its magnetic
moment) is equal to the energy of the "photons in the
electromagnetic field bathing the sample; most efficient
exchange of energy occurs when the nuclear energy levels are
in "resonance with the radiation field. The nuclear magnetic
energy depends on the local magnetic field rather than the
applied field, and these may differ because the applied field is
able to induce extra local fields in the vicinity of the nuclei.
For a given applied field nuclei in different chemical environ
ments experience different but characteristic local fields; there
fore they come into resonance with the radiation at different
values of the applied field. Different groups of nuclei therefore
give rise to absorption lines at different magnetic fields, and
since the n.m.r. spectrum is a plot of the absorption against
applied field, different molecules, and even different groups on
liqhfly shielded
heavily shielded
9806
low field
0?OT
■f rscale I
FIG. C1. The 60 MHz n.m.r. spectrum of acetaldehyde with TMS added
as reference. The S and the rscales for the chemical shift of GHO and
CH 3 are idicated. Note the fine structure.
36
chemical shift
the same molecule, give rise to absorptions in different regions
of the spectrum (Fig. CI). The separation of the absorption
lines is the chemical shift.
We deal first with the scales on which chemical shifts are
measured. The separation of the resonances is measured in
hertz {Hz, cycles per second), but, because the local field
depends upon the strength of the applied field, to quote a
separation is insufficient. Therefore the separation is divided
by the mean frequency, or, what is effectively the same, the
frequency of the stimulating field v . If one proton resonates
at a frequency v. and another at v., the value of 5.. = [v.— V.)/v
is quoted as the chemical shift on the 5scale. Since the
separation is generally no larger than about 1 kHz, and the
spectrometer operating frequency is of the order of 100 MHz,
the chemicalshift scale extends up to about 1Q~ S . It is con
venient therefore to express chemical shifts as parts per million
{p. p.m.), and so a value of 1 p. p.m. implies that two lines are
separated by 100 Hz in a spectrometer operating at 100 MHz,
and by 60 Hz in a spectrometer operating at 60 MHz. The
absolute chemical shift, the shift of the resonance of a proton
in a molecule relative to the bare proton, is of little practical
interest, and a scale often adopted sets the protons in
tetramethyl siiane (Me 4 Si, TMS) at the origin of the
chemtcalshift scale (6 = 0, by definition); heavily shielded
protons have small 5 values and resonate close to the TMS line;
lightly shielded protons have large 6 values and resonate well
to lowfield of the TMS line. Another scale is the Tsca/e, in
which TMS is set at T= 10; it follows that the scales are
related by r = 10—5. When the rvalue is large the nuclei are
heavily screened. Some representative rvalues are listed in
Table 4 on p. 268.
What are the contributions to the chemical shielding of a
proton? All the contributions may be ascribed eithento the
currents induced by the applied field on the atom itself or to
the currents induced in neighbouring groups. The currents
induced on the atom may be either paramagnetic or diamag
netic (see "magnetic properties), and these respectively
augment (deshield. move to smaller r) or oppose (shield, move
to larger r) the applied field. The general explanation of
'magnetic properties shows that the paramagnetic current is
small when the excited states of an atom are high in energy,
and this is the case in the hydrogen atom. For other atoms in
molecules there are several states that lie quite close to the
ground state, and in them the paramagnetic effects can domi
nate the diamagnetic. Diamagnetic currents are independent of
the availability of excited states and so always contribute to
the shielding (but might not dominate it).
When the currents are induced in a neighbouring group the
proton experiences an additional field which may be ascribed
to the induced magnetic dipole moment on the neighbour.
Thus if the susceptibility of the neighbouring group is Y„ in
the direction of the applied field, the latter induces a magnetic
moment X m B which is a source of field at the neighbouring
proton. Such a dipolar field has a (1— 3cos 2 f?)/ff 3 dependence
(see Fig. C2), and so the orientation of the proton and the
induced moment are of crucial importance in determining
whether the interaction is shielding or deshielding; the sign of
the shift also depends on the sign of the susceptibility.
Furthermore, for a tumbling molecule the localfield correc
tion disappears unless the magnetic susceptibility of the
neighbouring group is anisotropic, and so the shift is pro
portional to this anisotropy. Two famous cases are illustrated
C=C bond
shielded
shielded
b t 6
in diced rinij current
deshielded proton
FIG. C2. (a) Shows the regions of positive and negative chemical shift
in the neighbourhood of a double bond. If x,. and \. are the magnetic
susceptibilities parallel and perpendicular to the bond the shielding
constant varies as a~(x M  x, I tt3cos 2 0)/3R 3 . (b) Shows the ring
current contribution to the chemical shift of ring protons in benzene.
colour
37
in Fig. C2: the position of the resonance in the protons near a
C=C double bond is determined largely by the magnetic moment
induced in the TTelectrons of the bond. The effect of the field
this moment generates is to shield or deshield protons in the
vicinity, depending on whether they lie respectively inside or
outside a double cone of half angle 54° 44' (at which angle
1— 3cos 2 <? = 0). The other example is benzene, where the
proton shift to low field arises largely from the ability of the
applied field to generate a ring current when it is perpendicular
to the plane of the molecule.
The calculation of the chemical shift normally concentrates
on the calculation of the shielding constant. This relates the
local field to the applied field 8 through fi(local) ■ (1—0)6.
When a is negative the local field exceeds the applied field
(deshield ing), and the nucleus resonates at low field; when O is
positive the local field is less than the applied, the nucleus is
shielded, and the resonance occurs to high field. Just a little
care is needed to relate a positive or negative shielding constant
to a diamagnetic or paramagnetic current because the
neighbouringgroup effect depends, as we have seen, on the
orientation of the molecule, and, depending on the orientation,
either sign of the shielding constant can arise from either sign
of the current.
Questions. 1. What is the meaning of the term 'chemical shift'?
Is it dependent on the magnitude of the applied field? Is the
separation of the lines due to protons in different chemical
environments likely to be smaller or greater in a large applied
field than in a small field? What happens to the separation
when the field is changed from 14 kG to 50 kG? Is there any
advantage in doing n.m.r. spectroscopy at high magnetic
fields? What is the significance of the statements 'the protons
resonate at 5 TMS = 2' and 'the protons resonate at2r'? What
is the significance of a high rvalue? Put the following proton
resonances in order of increasing field: C$H 6 , CH 3 I, CH 3 OH,
CfiHu (see Table 4). What is the source of the differences
between the local field and the applied field? Under what
circumstances is there a deshield ing effect at the proton? What
is the role of the neighbouring group of a proton? Why is it
possible to get both positive and negative chemical shifts from
both paramagnetic and diamagnetic currents in the neighbour
ing group? The magnetic susceptibility of the benzene ring in a
direction perpendicular to the plane is — 9'5 X 10T 5 and parallel
to the ring it is —35 X 1CT 5 ; calculate the chemical shift at the
benzene protons using the formula in the caption of Fig. C2;
calculate the shift of a proton at a point 3 A above the centre
of the ring. What is the shielding constant in each case?
2. Derive an expression for the chemical shift of the protons in
benzene that takes into account the anisotropy of the mag
netic susceptibility of the ring and the free rotation of the
molecule, and show that the effect of the ring current would
vanish if the anisotropy were zero. Assume that the field
induces a magnetic dipole moment at the centre of the ring.
Further information. See MQM Chapter 1 1 for a detailed
account of some of the steps that are needed to calculate the
chemical shift of molecules in terms of the fields and currents
induced by applied fields. For an account of the role of the
chemical shift in n.m.r. see McLauchlan's Magnetic resonance
'OCS 1). Further details will be found in LyndenBell and
Harris (1969), Carrington and McLachlan (1967), Pople,
Schneider, and Bernstein (1959), Memory (1968), Emsley,
Feeney, and Sutcliffe (1965), and Abragam (1961). All these
books deal with the calculation and interpretation of n.m.r.
spectra in terms of the chemical shift; see also Further
information in "nuclear magnetic resonance. The calculation
required in Question 2 is done on p. 59 of Carrington and
McLachlan (1967). The nature of the ring current is discussed
by these authors and by Davies (1967).
colour. A material is coloured if it is able to absorb a band of
wavelengths from incident white light or if it is able to scatter
light of one frequency more effectively than another. If we
discount the blue of the sky the more common mechanism is
the former. Visible light spans the region from 700 nm (red)
through yellow (at about 580 nm), green (530 nm), blue
(470 nm) to violet (420 nm). The energies of the photons vary
from 1 7 eV to 30 eV over this range (see Table 5), and there
fore for a system to absorb in the visible it must possess
excited states within this distance from the ground state. When
red is absorbed (by virtue of the presence of lowlying excited
states) the object appears blue; when blue or violet is absorbed
it appears red. Lowlying energy levels are not particularly
common among living systems, and so the predominant
38
commutator
colours of naturally occuring substances tend to be towards
the red end of the spectrum. This is part of the reason for the
relative scarcity of purple cows, or bright blue dogs, plants, etc.
Furthermore, if living systems had lowtying excited states
they would be uncomfortably liable to photolysis. Chlorophyll,
however, is specially constructed to cope with this situation,
and so green is ubiquitous. The most intense colours are due
to "electric dipole transitions. A chromophore is a group whose
presence endows colour on a molecule. An important organic
chromophore is the C—C double bond, in which the relevant
transition is of an electron from the 7Tbond to the "antibond
ing TT*orbital (a 7T*<7r, 'pitopistar', transition). The carbonyl
group (C=0) is also important: the relevant transition is from
a nonbonding °lone pair on the oxygen to the 7r*orbital (an
'ntopistar', 7T**n, transition). This transition is electric
dipole forbidden {see 'selection rules and "oscillator strength),
and so is generally weaker than the ir**ir transition.
Transitionmetal complexes are often coloured: this is a
consequence of the presence of the delectrons and their small
energy splittings arising from the 'crystal field of the surround
ing ligands.The intensity of the colour is low because dd trans
itions are forbidden by the selection rules in operation, and
the intensity is due to a "vibronic transition. More intense
colours in the same systems (such as the characteristic intense
purple of permanganate, or manganate (VII) ion, MnO^) are
often due to chargetransfer transitions, in which electrons
migrate from the metal ion to the ligand or vice versa. Such
transitions possess a large "transition dipole moment.
Insulators are often colourless and, when crystalline,
transparent: think of diamond. This is because the electrons
are tightly bound to the atoms and are available neither for
conduction nor for light absorption (the conduction "band of
diamond can be attained only by supplying 5*3 eV of energy).
As the binding gets less tight, or as the conduction band
approaches the filled band, colour begins to appear: the band
gap in CdS 2 is 242 eV, and hence it is a yelloworange solid
(absorbing in the blue); silicon, with a band gap of only T14
eV, absorbs all frequencies and has a metallic lustre.
Metals are characterized by a shining surface when freshly
cut, and this hardly seems compatible with the remark that
they absorb all the incident light. The answer lies in the fact
that they also radiate all the incident light. This can be
envisaged in terms of the high mobility of the electrons in a
metal— an oscillating light wave approaches the surface, its
electric field drives the surface electrons back and forth, and
the incident light is quenched. But the oscillating surface
electrons themselves give rise to a radiated light field, and so
almost all the light is reflected. In the case of some metals
(copper and gold are familiar examples) there are true absorp
tion bands in the visible region, and both these metats extract
some blue light (and get hotter in the process).
Two examples of colouring arising through scattering may
be mentioned briefly. The sky is blue because blue light is
scattered more strongly than red; therefore more of the sun's
blue radiation is scattered down to us than is its red (except
in the late evening, when the sun appears ruddy because
someone further west is getting its blue light for his daytime
sky). The reason why clouds appear white even though their
presence is also seen by a scattering of incident light is to be
found in the size of the scattering particles (see Further
information). The other example is the classical colouring of
glass by the precipitation of colloidal gold: these minute
particles scatter away the blue component of transm itted
light, and to the glass there is imparted a rich ruby hue.
Further information. See Murrel! (1971) for an account of the
electronic spectra of organic molecules and Chapter 17 of
Kittel (1971) for a good survey of the optical properties of
insulators. A most pleasing account of the physical basis of
colour has been given in a simple article by Weisskopf (1968),
The photochemical aspects of the absorption of light are
described by Wayne (1970) and Calvert and Pitts (1966), on
which is based Table 5. The chemical aspects of light are also
described by Bowen (1946). The intensity of the absorption
of light depends, for "electric dipole transitions, on the
"oscillator strength, and experimentally it is expressed in terms
of the "extinction coefficient: see MQM Appendix 10.2 and
Wayne (1970). Concepts, methods, and data concerning
colour are described in Wyszecki and Stiles (1967).
Commutator. A commutator of two "operators A and B is
the difference AB—BA ; it is normally denoted [A,B] (AB
means that operation B is performed first, and is followed by
operation A ', BA implies that A precedes B.) Two operators
are said to commute if their commutator is zero. The non
complementarity
39
vanishing of a commutator of two operators indicates that the
final result of performing two operations depends on the order
in which the operations are done: operation A followed by
operation B and B followed by ,4 lead to different results.
For example, multiplication of a function f(x) by x followed
by differentiation is different from differentiation followed by
multiplication: (d/dx)xf(x) ^ x(d/dx)f{x). The commutator of
(d/dx) and x, [d/dx, x] , is 1 because from the rule for the
differentiation of a product (d/dx)xflx) is equal to
fix) + xWdxWW.
The importance of the commutator in quantum mechanics
lies in the theory's dependence on "operators: the manipu
lations of quantum mechanics must take the possible lack of
commutation into account; indeed, the very fact that commu
tators do not disappear is the feature responsible for the differ
ences between quantum and classical mechanics (see "matrix
mechanics). The "uncertainty principle applies to observables
whose operators do not commute.
The commutator of quantum theory is related to the
Poisson bracket of classical mechanics, and the recognition of
this connexion is reputed to have been the cause of one of the
most jubilant moments of Dirac's life.
The technical importance of the commutator lies in the fact
that the "eigenfunctions of one operator are also eigenfunc
tions of any other operator with which it commutes.
Questions. What is a commutator? Calculate the commutator
of (d/dx) and x, of d/dx and d/dy, of d 2 /dx 2 and x 3 , and of
x(d/dy) — K(d/dx) andz(d/dx) ~x(d/dz). Demonstrate the
validity of the following relations:
[A,B] = lB,Al
[A+B.C] = [A,C] + [B,CJ
[A.IB.C] ] + [B, [C.A] 1 + [C,[A.B1 ] = (Jacobi identity).
Prove that a necessary and sufficient condition for two
operators to have simultaneous eigenfunctions (that is, for the
eigenfunctions of one operator to be eigenfunctions of the
other) is that they commute.
Further information. See MQM Chapter 4 for a discussion and
proof of some of the consequences of the lack of commutation
of operators. The final Question is answered there (p. 108).
See the standard texts on quantum theory for a further
account; for example, Oavydov (1965), Landau and Lifshitz
(1958a), Messiah (1961), Schiff (1968), and Dirac (1958). For
deeper accounts see Jauch (1968) and Salem and Wtgner
{1972). The consequences of noncommutation for exponential
operators of the form expfiare deveJoped by Wilcox (1967).
Complementarity. The wave and corpuscular properties of
'particles' are complementary in the sense that an experiment
designed to determine the value of a wavelike property
automatically eliminates the precision with which a
corpuscularlike property may simultaneously be determined:
see 'uncertainty principle. Complementarity is the mutual
exclusiveness of these two types of property: it is impossible
to demonstrate simultaneously the wave and corpuscular
attributes of a particle.
Further information. See "duality, "uncertainty principle, and
"wave packet for a more detailed discussion. See p. 1 58 of
Bohm (1951), Kramers (1964), and Jammer (1966).
Compton effect. Light scattered from electrons shows an
increase in wavelength which ts independent of its initial wave
length but characteristic of the angle through which it is
deflected.
On the basis of classical theory it is surprising that only one
value of the wavelength shift is observed for a particular angle
of deflection, and the result strongly suggests that a collisional
process is involved. If it is assumed that a "photon of energy bv
and momentum A A is in collision with a stationary electron, and
that both energy and momentum are conserved in the collision
of the two particles, then it is a simple matter to deduce the
expression 5X= \falm t)(1— cos 9). SAisthe wavelength shift
(always an increase), and is the deflection of the light. This
expression is indeed independent of the initial wavelength, and
gives a unique value of SXfor a given 6. The agreement shows
the essential validity of the collisional model, and so it is
excellent evidence for the quantization of light into "photons,
and for their behaviour as particles.
The quantity b/m c is the Compton wavelength A and its
numerical value is 0024 A, or 24 pm (to be precise:
2'426 309 6 X 10~ 12 m). Therefore even in the backward
40
configuration
scattering direction [0 = 180°} the wavelength shift is only
48 pm, and this smalfness indicates why it is necessary to use
Xrays or 7rays, for only then is the shift a significant pro
portion of the wavelength: the effect is independent of the
wavelength, but it is easier to detect.
Questions, 1. What are the characteristics of Compton scat
tering? What features are inconsistent with a classical view of
the process as an interaction of a charged particle with an
electromagnetic wave? Why is the effect unimportant at large
wavelengths? Calculate the Compton wavelength of a proton.
2. Deduce the Compton formula. Take an initial photon wave
length A and a final wavelength X+ 5X; let the electron be at
rest initially and after the collision have a kinetic energy ^mv 1
and momentum mv (do everything nonrelativistically). Write
the expression for the conservation of energy during the
collision (it is an elastic process), and then do the same for the
linear momentum on the basis that the light is deflected
through 8, Expand the quantities to first order in SK and
eliminate v to get the final expression.
Furtfter information. See §2.8 of Bohm (1951 ) for a discussion
of the differences between the classical and quantum situations
and Chapter 4 of Jammer (1966) for an historical perspective.
The original work is described by Compton (1923). More com
plicated treatments of the Compton effect are to be found in
§22 of Heitler (1954), Chapter 1 1 of Jauch and Rohrlich
(1955), and Schweber (1961). Note that the nonrelativistic
limit of Compton scattering is normally called Thomson
scattering, and the KleinNishina formula was the result of the
first calculation of the crosssection for relativistic photon
electron scattering. These matters are discussed in the cited
references.
configuration. The electronic configuration of an atom or
molecule is the description of the way the electrons are
distributed among the available orbitals. Thus the configur
ations of the first row atoms are H Is; He Is 2 ; Li Is 2 2s;
Be 1s 2 2s 2 ;B ts 2 2s z 2p, and so on up to Ne 1s 2 2s 2 2p*. Some
times the inner complete shells are abbreviated to K,L,M, etc..
Thus the ground configuration of sodium could be written
eitheras1s 2 2s 2 2p 6 3soras [KL] 3s. In a similar fashion the
configurations of molecules may be written in terms of the
occupied orbitals. Thus we have H 2 1sa 2 ;0 2 IsfJ 2 Iso" 2
2so 2 2so* 2P0 2 2plf 2p7r* 2 , and the latter is often
abbreviated by the weak but wise to . . . o^Tr 2 , If the else
9 u g
tronic configuration is known {for example, by applying the
"aufbau principle) the spectroscopic "terms may be deduced
and spectra predicted.
Questions. What is a "configuration' of an atom? What may be
deduced if the configuration is known? Write down the con
figurations of the atoms He, B, C, O, F, Al, Si, CI, Cu, Fe, U,
and Cf in an economic fashion. Do the same for the molecules
H 2 , HD, N 2 , f%, Ne 2 , FeO, and "benzene.
Further information. The standard collection of atomic
energy levels, which perforce contains lists of configurations,
is that of Moore (1949 et seq.). Standard works on atomic
structure are those of Herzberg (1944) and Condon and
Shortley (1963). Candler (1964) is helpful. See also MQM
Chapter 8, the synopsis of atomic spectra, and the "aufbau
principle.
configuration interaction. A xonfiguration tells us how
electrons are distributed among the available atomic or
molecular orbitals, and the simplest description of the struc
ture of an atom or molecule consists of a statement of its
configuration. Thus molecular hydrogen could be described as
having the structure '\so 2 . The energy of the molecule corre
sponding to this configuration could then be quoted as the
'molecular energy'. Such a description might not be particu
larly good, for a true description of the molecule ought to
allow for the contamination of the configuration by some
others: we know that a molecular wavefunction is inaccurate
for a variety of reasons, and so it is certain that the single
configuration description of its structure will be inaccurate in
some fashion. For example, the singleconfiguration descrip
tion of the molecule of hydrogen is deficient in as much as it
permits both electrons to be localized on the same nucleus to
too great an extent.
The situation can be improved by modifying the wave
function by permitting configuration interaction (CI): we
permit the wavefunction of the molecule to be described by a
mixture (a •superposition) of wavefunctions corresponding to
different configurations. Admixture of some excited state con
conserved property
41
figurations distorts the groundstate function, and if the
mixture lowers the energy of the ground state then the
■variation principle states that we have an improved descrip
tion of the molecule. In the case of H 2 it turns out that a
major improvement in the energy, and therefore the wave
function, can be brought about by mixing in some of the
configuration tsCT* 2 : when the effect of this admixture is
analysed {see Questions) it emerges that the effect of allowing
configuration interaction is to reduce the contribution of
situations in which both the electrons are at the same nucleus.
Those who like the language of electron correlation theory
will realise that CI has achieved a certain amount of charge
correlation: simple molecularorbital theory underestimates
electron correlations, and the admission of CI goes some way
to repair the defect.
What configurations can be mixed into the ground state?
The first requirement is that they have the same symmetry:
for example, H 2 is improved by the admixture of '2 rather than
3 £or 1 I1 The second requirement stems from "Brillouin's
theorem, and is that singly excited states do not interact
directly with the ground state if proper selfconsistent field
configurations are being considered.
The effect of CI may extend beyond improvement to the
molecular energy, because the fact that excitedstate con
figurations are mixed into the ground state means that the
ground state of the molecule possesses some of the character
istics of these excited states. Such contamination may
influence the predictions of the °Hund rules, for instance.
The simplest way of doing a CI calculation in practice is to
calculate the orbitals and their energies, and then feed in
electrons to form various configurations. The ground state
corresponds to the configuration with the lowest energy. Then
the actual wavefunction of the molecule is expressed as a
•linear combination of these configurations {using the sym
metry criteria to decide which configurations one should
bother about), and then the variation principle is used to
determine the best mixture. A better, but much more com
plicated, procedure is the multkonfigurational calculation in
which the best structure for each configuration is calculated
separately {rather than by applying the "at/fbau principle to
a single set of levels), and then the variation principle is
applied to these optimized components.
Questions. 1. Why is a singleconfiguration description of a
molecular state possibly a poor description of its actual struc
ture? In what respect is a singleconfiguration description of
the hydrogen molecule a poor object in simple molecular
orbital theory? What is configuration interaction, and why
does it overcome some of the defects of the single
configuration method? What effect does it have on the inter
pretation of the wavefunction of the ground state of Hj?
What other influences on molecular properties may CI have?
Suggest the influence of CI on the stability, dipole moment,
magnetic susceptibility, and polarizability of molecules. Will
spectral selection rules be modified by CI? What are the
criteria for selecting the configurations able to interact with
the ground state?
2. The energy of a configuration A ties at Af above the ground
state configuration of energy E . Suppose that there is an
interaction between the configurations (that is, there is a
matrix element V between the configurations). Use the
variation principle to show that the revised groundstate
energy with CI is given by the smaller root of a quadratic
equation (obtained from the "secular equations). Make a rough
estimate for Af in H 2 and use a value of V = 1 eV to compute
the modifications to the curves on CI. Is it reasonable to use
the same value of V at all internuciear distances? What is the
source of the interaction responsible for the magnitude of V?
3. Extend the results of Question 2 to compute the wave
functions of the two states. Show that the configuration o 2 ,
with u proportional to 1s g + 1s b , is modified by the admixture
of the configuration a* 2 , withe* proportional to 1s a — 1s b , in
a way that can be interpreted as an improvement of the dis
tribution of the electrons. Discuss the role of CI on the
products of dissociation of the H 2 molecule.
Further information. See MQM for a short discussion of mole
cular hydrogen in terms of CI, and see also Coulson (1961). A
simple pragmatic approach to the subject will be found in
Richards and Horsley (1970), and a lengthier, but helpful,
account is given by McGlynn, Vanquickenborne, Kinoshita,
and Carroll (1972); both work through some examples.
conserved property. A conserved quantity or property is
one whose value does not change with time. A familiar
42
correlation energy
example is the total energy of a system (First law of thermo
dynamics); another is the angular momentum of an electron in
an atom, or the component of angular momentum about the
internuclear axis in a diatomic molecule. An example of an
unconsented quantity is the angular momentum about an axis
perpendicular to the internuclear axis in a diatomic molecule:
this can be envisaged as the electron beginning its journey
about the perpendicular axis, but colliding with a nucleus
before it has completed its rotation. This collision changes the
value of the angular momentum about the perpendicular axis,
and so it is not a conserved quantity. A conserved quantity is
also called a constant of the motion.
The definition of a conserved property can be made
quantitative by defining it as an observable whose corres
ponding "operator "commutes with the "hamiltonian of the
system. For example, the linear momentum of a system is
conserved if the linear momentum operator commutes with
the hamiltonian for the system. In the Questions you are asked
to show that the commutator of the linear momentum and
the hamiltonian is proportional to the gradient of the potential
energy of the system. From classical mechanics we know that
the gradient of the potential is a force, and so we arrive at the
pleasing conclusion that the linear momentum is a conserved
quantity only in the absence of a force. Alternatively, we may
conclude that in the presence of a force the linear momentum
is not conserved, or that the linear momentum changes. This is
essentially the content of Newton's second law of motion.
Questions. What is meant by the term 'conserved property'?
Give some examples other than those mentioned in the text.
Consider the "expectation value of the 'operator corres
ponding to some observable £2. By invoking the time
dependent form of the "Schmdinger equation, show that the
rate of change of the expectation value, (d/df) {£2) , is equal
to (i/li) <[A/,J2] >. This shows that if the commutator of H and
£1 disappears, then the expectation value of £2 remains
constant in time. Now let £2 be the linear momentum in the
^direction and let H be the operator composed of the
"kinetic energy plus some potential energy V(x). Deduce an
expression for the rate of change of the expectation value of
the linear momentum in terms of the expectation value of a
force. Recognize and, if your spirit is of that kind, be thrilled
by the fact that you have deduced Newton's second law of
motion. Reflect on it by considering the fact that, in the light
of this result, Newton's law may be regarded as an equation for
average values, and so realize that classical mechanics is a
treatment of averages and so ignores the finer details of
motion. What are the conditions that must be satisfied in
order that energy, angular momentum, and linear momentum
bo conserved in mechanical systems?
Further information. See MQM Chapters 4 and 6 for a short
discussion of conservation laws. More detailed information
will be found in Chapter 5 of Roman (1965), and Feynman,
Leighton, and Sands (1963). On p. 195 MQM discusses the
fact that the conservation of energy, momentum, and angular
momentum are features of the symmetry of space.
correlation energy. If it is well done, a "selfconsistent
field calculation of an atomic or molecular structure in the
HartreeFock scheme generally gives a good answer; but even
if the calculation is done exactly within the scheme the cal
culated energy differs from the true energy. A part of the
discrepancy lies in the neglect of relativistic effects, which
might be very large for innershell electrons possessing a high
kinetic energy; but even when this is allowed for there remains
a discrepancy. The magnitude of this residual difference is the
correlation energy, and its presence reflects the approximations
inherent to the HartreeFock scheme. A basic approximation
of the HartreeFock scheme is the neglect of the local distor
tion of the distribution of electrons, and the averaging of its
effect over the whole orbital: instead of an electron's orbital
being distorted in the vicinity of another electron, the whole
orbital is modified in an averaged way (Fig. C3). Therefore
the scheme neglects local electronelectron effects— it neglects
electron correlations.
The effect of the neglect of correlations is to cause the
calculated molecular potentialenergy curve (see Born
Oppenheimer approximation) to lie above the true curve.
Nevertheless, the true shape is approximately reproduced,
except that it is too narrow in the region around the minimum
(the equilibrium bond length). The minimum of the Hartree
Fock curve occurs at about the same position of the true
curve. Thus the equilibrium geometries are quite well predicted
correspondence principle
43
firsf qu«s
first <fieu
Hcirrreefbck
corrected function
FIG. C3. (a) The actual distortion and (bl the HartreeFock pretence.
if correlation effects are neglected, but the "forceconstants,
and hence the molecular vibration frequencies, are exagger
ated.
Questions. Even the best HartreeFock calculation does not
give the exact electronic energy of an atom or molecule: what
are the reasons? What is the correlation energy, and why is it
so called? How must the HartreeFock scheme be improved in
order to regain more accurate results? If a molecular potential
energy curve is calculated on the basis of the HartreeFock
scheme, in what way should it be expected that the equilib
rium bond lengths and vibration frequencies will differ from
their actual values? Would you expect the neglect of corre
lation effects to be more or less important as the bond between
two atoms lengthens? What features would you think of incor
porating into a wavefunction of the helium atom in order to
reflect the tendency of the electrons to remain apart?
Further information. A simple account of the methods that
are employed to deal with the calculation of atomic and mole
cular structures, and the role of correlation effects, is con
tained in Richards and Horsley {1970} and in McGlynn,
Vanquickenbome, Kinoshita, and Carroll (1972). For some of
the ways the problem has been tackled, see Pauncz (1969),
Berry (1966), Sinanoglu (1961 ), Clement! (1965), and
Sinanoglu and Brueckner (1970).
correspondence principle. At large "quantum numbers the
mean motion of a system becomes identical with its motion
calculated on the basis of classical mechanics.
The principle implies that the rules of quantum mechanics
contain the structure of classical mechanics when the fine
details of the situation are ignored. An example of its appli
cation is provided by the construction of a 'wave packet to
represent the motion of a free particle when the energy or
momentum is only coarsely specified: the packet moves
along the same trajectory that the same mass point would have
in classical mechanics. But as the energy becomes more
precisely specified, so that fewer quantum states contribute to
its representation, the distribution becomes less classical and
more quantal. Another example is provided by the Planck
distribution of energy in a "blackbody radiator: as Planck's
constant dwindles to zero (in a hypothetical classical world)
the energy distribution becomes that of a classical system and
agrees with the R ay leigh Jeans distribution law. Likewise, the
momentum of a "photon (a light quantum) is transmitted to
the object that absorbs or reflects it; and when a sufficiently
large number of photons is involved this impulse is interpreted
as the steady radiation pressure of classical electromagnetic
theory.
The correspondence principle was of profound importance
in the early days of quantum theory for it acted as a guide
through the exciting gloom of those days: any calculation
based on quantum theory had to correspond to a classical
result in all details when sufficiently large quantum numbers
were involved and quantum fluctuations ignored. As an
example of this kind of development we may consider the
radiation emitted by an harmonically oscillating electron: the
existence of an array of equallyspaced quantized energy levels
in a "harmonic oscillator suggests that a very wide range of
frequencies could be emitted because of the indefinitely large
number of different energy separations that may be obtained.
Nevertheless, a classical oscillator, to which the quantum
harmonic oscillator must correspond, emits only a single
frequency, that of its natural classical motion. In order to
reproduce this result it is necessary to impose restrictions at
the quantum level on the transitions that can occur; thus the
correspondence principle leads to the "selection rule that an
oscillator may make a transition only to a neighbouring level.
44
Coriolis interaction
Such rules emerge naturally from the later quantum mechanics,
but even there the correspondence limit is often a very good
check on the validity of a calculation. Finally it should be
noted that purely quantum phenomena disappear in the
correpondence limit; in particular, all effects due to °spin are
eliminated.
Further information. Various classical limits of quantum
situations are discussed in MQM Chapter 3. For an historical
perspective and an account of the way the principle was used to
disentangle the old and discover the new quantum theory see
Jammer (1966). Applications of the principle are described in
Kemble (19581 and Kramers (1964).
Coriolis interaction. The Coriolis interaction is the inter
action between the rotation and vibration of a molecule.
Think first of a diatomic molecule rotating about an axis
perpendicular to its internuclear axis, if we regard it as a
classical problem we may imagine the bond lengthening and
shortening as it vibrates, and this vibration changes the
moment of inertia of the molecule. 'Angular momentum is
conserved for the molecule, and so for the same angular
momentum, but smaller moment of inertia /, in order to
preserve the product lio it is necessary for the angular velocity
toto increase. Conversely, as the bond lengthens and the
moment of inertia increases, the angular velocity must fall in
order to conserve the angular momentum. Therefore the
rotation accelerates and decelerates as the bond vibrates. (This
is the same mechanism whereby managing directors on ro
tating chairs, and ice skaters, speed or slow their rotational
motion.) This picture shows that there is an interdependence
between the rotation and the vibration of a molecule, and it
may appear in modifications of the spectral lines.
An important application of the Coriolis interaction is
encountered in the case of a vibrating, rotating, linear, tri
atomic molecule. There are four "normal modes of vibration
of this molecule; one is a symmetric stretch in which both
A— B bonds vibrate in phase, another (named l> 2 ) is the
asymmetric stretch in which as one A— B bond shortens the
other lengthens, and the other two are the bending motions
which may occur in two perpendicular planes {see 'normal
modes for pictures). Consider the effect of the v 2 vibration
FIG. C4. The asymmetric stretch (labelled v 2 1 and the molecular
rotation interact and induce a bending vibration in the plane o!
rotation.
interacting with the rotation by the Coriolis mechanism
(Fig. C4). As one A— B bond shrinks there is a tendency for
that half of the molecule to speed up; therefore that bond
tends to bend forward relative to the rest of the molecule.
Meanwhile the other bond is getting longer, and the Coriolis
interaction requires that half of the molecule to decelerate;
therefore it tends to lag behind in the rotation of the molecule.
The net effect is that the molecule tends to bend in the plane
of the rotation. But having got this far the stretching motion is
at the end of its swing, and it begins to swing back: the long
bond shortens and the short lengthens. The Coriolis interaction
comes into operation and the faster rotating bond becomes the
laggard and vice versa. This induces the molecule to bend the
other way. The net effect of the continuing process is that
the bending vibration is stimulated by the combined anti
symmetric stretch and the rotation of the molecule. This
means that the two vibrations are not independent and that
Coulomb integral
45
the effect of the Coriolis interaction has been to mix different
vibrations together. This appears in the spectrum as £type
■doubling.
Questions. What is the Coriolis interaction? Why is it able to
cause an interaction between the vibrational and rotational
motion of a molecule? Discuss the changes brought about by
the change in the moment of inertia of a molecule by virtue of
the letter's vibration on the angular speed, frequency,
momentum, and rotational energy of the molecule. Discuss the
effect of the Coriolis interaction on the vibrations of a linear
triatomic molecule. What is it unable to mix? What is the
consequence of the mixing on the spectrum? Apply the same
reasoning to a planar triangular molecule of the form AB 3 .
Which vibrations of this molecule is the rotation able to mix?
Consider the effect of rotations about the axis perpendicular
to the plane, and about an axis in the plane of the molecule.
What other manifestations of the Coriolis force can be ob
served in macroscopic phenomena?
Further information. See MQM Chapter 10. The classical basis
of the Coriolis interaction, which is concerned with the way
mechanical systems behave in rotating axis systems [rotating
frames) will be found in §4.9 of Goldstein {1950). Appli
cations to spectroscopy will be found in King (1964),
Herzberg {1945, 1966), Sugden and Kenney (1965), Wilson,
Decius, and Cross {1955), and Townes and Schawlow (1955).
A helpful theoretical discussion is to be found in Allen and
Cross (1963).
Coulomb integral. The Coulomb integral is the contribution
of the classical electrostatic interaction between charge dis
tributions to the total energy of an atom or molecule; although
there are some frills which must be added to this basic de
scription.
In atoms the frills are fewer and we deal with them first.
Consider an electron distributed in an "atomic orbital \p :
a
knowing the mathematical form of the orbitals we are able to
say that in the small volume element dr, surrounding the
point r, the probability of finding the electron is \\p {r^PdTi ;
therefore the charge in that region isdgi = el^ a (ri)rdri.
Another electron present in the atom may occupy the orbital
t// ; by the same argument the charge in some little volume
element dT 2 surrounding the point r 2 is dg^ = — e\\j/ .{rjlPd^.
If the charges 6qi and dg 2 are separated by a distance r 12 the
potential energy of their interaction comes from the expression
for the Coulomb potential, namely dq l dq 2 /$'n€f ) ri2. It follows
that the two charges at t x and r 2 will give a contribution of
this form to the total energy of the atom. The total contri
bution to the energy can be obtained by summing over all the
volume elements dT[ and drj, or, since these are infinitesimal
volume elements, by integrating over them. The procedure leads
to the Coulomb integral J as illustrated in Fig. C5a. As the
charges are the same, J is positive, and so leads to an increase
in the energy of the atom. This means, of course, that the
Coulomb interaction is repulsive. The frills to tack on to this
description are concerned with the intrinsic tendency of
electrons to correlate their motion: the correction to the
Coulomb energy arising from this effect is termed the
"exchange energy.
J =^T,d Tj \% Cr^nvJ %('$
nner shells
FIG. C5, The contributions to the Coulomb integral (a) in an atom,
lb) in the VB theory of a diatomic molecule, and Ic) in MO theory of
(he same. Arrows like "*~* represent attractive interaction, and  *  *■
repulsive.
46
crystalfield theory
In molecules the situation is complicated by the variety of
interactions possible when several nuclei are present. It is
further complicated by the existence of two different de
scriptions of the chemical bond: the °molecu!arorbital and
the valencebond theories. Nevertheless, in each one the term
'Coulomb integral' signifies the type of energy that arises by
virtue of the classical electrostatic interaction between charges.
In the valencebond method the Coulomb integral is
composed of three parts: the electrostatic repulsion between
the two nuclei in the bond, the repulsion between the charge
distributions on the two nuclei, and the attraction between the
charge distribution on one nucleus and the opposite charge of
the other nucleus {see Fig. C5b). This integral is also called J,
but we note that it may be negative if the last of the three
contributions exceeds the others. It too must be corrected to
allow for electron exchange: details are given under "valence
bond.
In the molecularorbital method the Coulomb integral,
which especially in the "Hu'ckel method is often denoted a,
generally consists of four parts. The first is the energy of the
electron as if it were in an orbital of the isolated atom (this
therefore contains the attraction of the electron to its own
nucleus, its repulsion from the other electrons on the atoms,
and its kinetic energy); the second is the nucleusnucleus
repulsion in the bond; the third is the electrostatic attraction
between the charge distribution and the neighbouring nucleus;
and the fourth is the repulsion of the electrons on the two
nuclei. These interactions are illustrated in Fig. C5c and dis
cussed further in the entry on "molecularorbital theory. It too
must be corrected for "exchange interactions.
The feature common to all these Coulomb integrals is the
way they select, from all the interactions within the molecule,
the ones that would be selected on a simple classical electro
static picture of the interactions; they ignore the way that the
'spin of the electrons correlates their motion.
Questions. What is meant by the expression 'Coulomb
integral'? How does the Coulomb integral appear in the theory
of atoms: present a simple argument on the basis of the
expected electrostatic interactions, and demonstrate how the
expression emerges from the proper "hamiltonian for the
energy of a manyelectron atom. Express the energy of the
helium atom in terms of the Coulomb integral J. Can J (for
atoms} ever be negative? What additional complications are
there in molecules? What are the components of the
electrostatic energy in the valencebond description of a
chemical bond between two atoms? What are the correspond
ing contributions in the molecularorbital description of the
same bond? What features are common to both? What
features are unique to each? May the molecular Coulomb
integrals be negative? What would that signify? What additional
contributions to the Coulomb integral might be expected when
a molecule containing more than two nuclei is being con
sidered? What is the maximum number of different atomic
orbitals that need be considered inside a Coulomb integral?
Further information. See MQM Chapters 8 and 9 for the role
of the Coulomb integral in atomic and molecularstructure
calculations. See also Coulson (1961). A helpful account of
the calculation of Coulomb integrals will be found in McGlynn,
Vanquickenborne, Ktnoshita, and Carroll (1972). Applications
to the determination of molecular structure will be found
there and also in Richards and Horsley (1970). See the entries
on "exchange energy, "spin correlation, "selfconsistent fields,
and the "molecularorbital and "valencebond theories.
crystal field theory. The five dorbitals of a free transition
metal ion are "degenerate (have the same energy), but when
the ion is complexed the ligands remove the spherical
symmetry of the atom and replace it by a field of lower
O "
,
■6Dq /
1 /
44
IfjDq
4Dq \
1 V
■
r 2q WxyV'W
FIG, C6. The effect of an octahedral crystal field on the energy ol an
electron in the dorbitals of the central atom.
crystalfield theory
47
low spin
M^"
1
strontj fiefd
weak fieH
to field
FIG. C7. The competition between spinpairing and 100<j leading to
lowspin or highspin complexes,
symmetry. The immediate environment of the ton may be
octahedral, tetrabedral, and so on. The ligands may be
regarded as a source of electrical potential, the crystal field,
and the dorbital energies adjust accordingly. In an octahedral
complex the d 2 and d 2 2 orbitals point directly towards
the ligands (Fig. C6), whereas the other three dorbitals point
in mutally equivalent directions between them. It follows that
the former pair possess one energy and the latter trio a differ
ent energy, and so the degeneracy of the orbitals has been
removed. The energy separation is conventionally called \0Dq.
Electrons may be fed into the array of atomic orbitals so
formed, and the structure, spectra, stability, and magnetic
properties of the complex discussed on the basis of the 'aufbau
process. The competition that governs the configuration of an
ion in a complex is between the tendency of the electrons to
enter the lowestenergy orbitals, which more than three can
do only if their spins pair, and the tendency for electrons to
enter a set of orbitals with parallel spins ("Hund rules). Which
tendency wins is governed largely by the separation of the two
sets of orbitals, and therefore by the strength of the crystal
field. A highspin complex results when the tendency not to
pair wins, and a lowspin complex results when the electrons
achieve lowest energy by pairing and entering the lowlying
orbitals (Fig. C7).
The crystalfield theory is an approximation because it
supposes that the energy of ligandmetal bonding is due
solely to electrostatic effects, and ignores both the covalent
nature of the bonding and the role that TTbonding might be
expected to play: these deficiencies are repaired in the
broader "ligandfield theory. Nevertheless, the crystalfield
theory is successful in so far as it expresses the importance of
the symmetry of the complex in determining the electric,
magnetic, and chemical properties, and provides a simple rule
for predictions of these properties.
Further information. See MQM Chapter 9 for a more detailed
account. An introduction to the idea of the crystal field is
provided by Earnshaw and Harrington in The chemistry of the
transition elements (OCS 13). A pleasing introduction to
crystalfield theory is provided by Orgel (1960), developed by
Ballhausen (1962), and consummated by Griffith (1964). See
ligand field for further directions.
D
de Broglie relation. According to the de Broglie relation a
particle travelling with a linear momentum p has associated
with it a wavelength A ■ hip. As the particle's momentum
increases the associated wavelength decreases. The relation can
be understood in terms of the interpretation of the 'wave
function of the particle, and the connexion between the
function's curvature and the particle's "kinetic energy. As the
kinetic energy of the particle increases, the curvature of its
wavefunction increases; but increasing curvature implies that
the wave swings from positive to negative amplitude more
often in a given length: its wavelength decreases. But the
kinetic energy is proportional to the square of the momentum,
and so the inverse relation between momentum and wave
length emerges.
A more direct demonstration of the quantummechanical
basis of the de Broglie relation is to observe that the wave
function for a particle with momentum p = ki\ is exp{\kx). If
this is written as CQ${2~npx/h)+\s'm(2Tipxlh) it can be seen
immediately that the function corresponds to a superposition
of real waves each with wavelength hip.
Questions. What is the de Broglie relation? How can it be
justified quantum mechanically? Why does the wavelength of a
particle decrease as its momentum increases? How could the
existence of the de Broglie wave be demonstrated experimen
tally? Calculate the de Broglie wavelength for the following
particles: an electron accelerated from rest by a potential of
10 V, 1 kV, 100 kV; a proton travelling at 1 km s _1 ,
1000 km s~';a mass of 1 g travelling at 100 km h" 1 ; a car of
mass 1 500 kg travelling at 50 km h  ' . Which of these species
would be expected to show pronounced quantummechanical
behaviour?
Further information. See Feynman, Leighton, and Sands
(1963} and Chapter 3 of Bohm (1951). For an account of the
de Broglie relation by two students of de Broglie, and an
account of the way that some people are attempting to inter
pret it, see Andrade e Silva and Lochak (1969). A pleasing
account of de Broglie's contribution to quantum theory is
given in §5.3 of Jammer (1966).
degeneracy. When two or more different "wavefunctions of a
system correspond to the same energy they are said to be
degenerate. Thus the three /7porbitals of any free atom con
stitute a triply degenerate set of functions, and the TTorbitals
of diatomic molecules are doubly degenerate. When only one
wavefunction corresponds to a particular energy the state is
said to be nondegenerate ('singly degenerate' is sometimes
used in this case, but this seems to be an illicit extension).
'Razing the degeneracy' of a state means removing the
degeneracy (for example, by applying an electric field to an
atom).
The degeneracy of a system is related intimately to its
symmetry, and whenever a wavefunction can be changed into
another by a "symmetry operation the functions are degener
ate. Thus rotating a free atom through 90° about z is a
symmetry operation, and as it rotates an np orbital into an
np orbital these are degenerate. In a similar fashion np^
may be generated by rotation about another axis, and so it too
is degenerate with np (and np ). A 3porbital cannot be
k y
48
density matrix
49
generated by rotating a 2porbital, and so, according to the
symmetry definition, they are not degenerate in atoms— and
this accords with common sense.
When two functions correspond to the same energy by a
numerical coincidence and not by virtue of the existence of a
symmetry operation they are said to be accidentally degener
ate. As an example, it is possible that the 2sO"* and the 2pJT
orbitals of some diatomic molecule have the same energy to a
few significant figures; they are then accidentally degenerate.
A widely quoted example of accidental degeneracy is that of
the "hydrogen atom, in which all the orbitals of a given
principal quantum number (that is, thews, Dp, nd, , . , orbitals)
have the same energy; but this is a fallacious example because
it is possible to find a cunningly hidden symmetry operation
by which it is possible to rotate an /jsorbital into an nporbital,
etc., and so the degeneracies are true rather than accidental.
The same 'hidden symmetries' can account for the apparently
accidental degeneracies of a "particle in a box whose sides are
in rational proportion (see Question 2), and all exact 'acciden
tal' degeneracies can be explained in terms of a deeper scrutiny
of the symmetry of the system.
Questions. 1 . When does a degenerate state occur? How is
degeneracy related to the symmetry of a system? What is
accidental degeneracy? How manyfold degenerate are the
ground states of the sodium atom and the boron atom {neglect
spinorbit coupling)? What is the degeneracy of the "hydrogen
atom when the electron occupies a state with principal
quantum number 1, 2, n? What is the degeneracy of the spin
state of an electron in zero magnetic field, and what is its
degeneracy when the field is applied? What can be said about
the degeneracy of an electron in the dshell of a transition
metal ion when an octahedral ligand field is present? Discuss
the last problem by symmetry arguments.
2. Take the energy levels for a "particle in a twodimensional
rectangular well (Box 15 on p. 166) and consider the case
where the sides are equal. Show that the lowest energy state is
nondegenerate, and that the next highest state is doubly
degenerate. Show that the two wavefunctions for the first
excited state can be interrelated by a rotation of the square
through a rightangle. Can any of the states be triply degener
ate? Can any state of a cubic box be triply degenerate? Now
let the well have sides a and ca. where c is a rational number:
find some degenerate states. Are these truly degenerate or
accidentally degenerate? Is there a hidden symmetry? What
happens when c is irrational?
Further information. The best way of discussing degenerate
situations is in terms of "group theory: see MQM Chapter 5,
Cotton (1963), Bishop (1973), and Mcintosh (1971). For a
discussion of the hydrogen atom degeneracy see MQM
Chapter 3, Englefield (1972), Mcintosh (1971), and Bander
and Itzykson (1966). The razing of degeneracy is the basis of
the crystalfield description of transitionmetal ions; therefore
read Earnshaw and Harrington's The chemistry of the
transition elements {OCS 13), Orgel (1960), Ballhausen
(1962), and Griffith (1964). Accidental degeneracy is dis
cussed in detail by Mcintosh (1958, 1971).
density matrix. In elementary applications of quantum
mechanics calculations are based on the "wavefunction \jj; but
whenever an actual observable is calculated one encounters
formulae which involve the wavefunction as its square \j/*\p
or in a bilinear combination of the form ]b*j!/ ; see, for
example, "expectation value or "transition probability. Why
not set up a formulation of quantum mechanics that deals with
the bilinear combinations \b*}jj directly rather than introduc
ing them only when an observable is being calculated? Further
more, the absolute phase of the wavefunction is immaterial,
for if both i^ m and $ are multiplied by the arbitrary phase
factor expi0 the product i^* 1^ remains unchanged, and so all
observables are independent of the absolute phase; it appears
that in dealing with the wavefunction one is carrying around a
piece of useless information. The final preliminary point to
make is that the state of a system is only very rarely pure:
normally an actual system has to be treated as a statistically
large collection of subsystems, and the result of an experiment
is determined by some average (actually one of the ensemble
averages of statistical mechanics) of the products 4l* 4> ; this
* — i —
statistical average we denote \p m \p n , and interpret the averaged
product as the n, m element of a matrix, the density matrix p.
The "matrix of elements p , which are by definition &*\}j ,
should contain all the information about the system: it carries
all the pertinent information of the wavefunction, and all the
50
dipole moment
information about the role of the statistical averaging. It is
not difficult to find an equation of motion of the density
matrix (corresponding to the "Schrodinger equation for the
wavefunction), but it is often quite difficult to solve. Never
theless, powerful methods exist for dealing with the density
matrix, and it is a common way of performing calculations
dealing with the timeevolution of complex systems.
Further information. For a gentle introduction, see Chapter
5.4 of Slichter (1963). Then see Chapter 4 of Ziman (1969),
Chapter 1.8 of Roman (1965), and Fano (1957). For
applications to molecular and atomic structure, see McWeeny
and Sutcliffe (1969).
dipole moment. A positive (+t?) and a negative (— q)
charge separated by a distance R constitutes an electric dipole
moment fi = qR. On a molecular scale a separation of the
order of 1 A (01 nm) is a typical magnitude, and the charge
of the electron a typical charge (480 X 10~ 10 e.s.u.,
160 X 10" l9 C); therefore molecular dipole moments should be
expected to have a magnitude of approximately 10" 18 e.s.u. cm
(3*3 X 10" 30 C m). Dipoles do indeed have this size in many
systems, and so the unit debye (1 D = 10" 18 e.s.u. cm,
33 X 1Q~ 30 C m) is frequently used when magnitudes are
quoted. Some typical molecular dipole moments are listed in
Table 6. A dipole moment is a directed quantity because at
one end is a positive charge and at the other a negative; in
order to denote the direction of the dipole in a molecule the
convention is employed of representing it by an arrow with
the head at the negative end: H — ► — . Expressions for the
energy of a dipole when it is in an electric field, and related
information, are recorded in Box 5.
The vectorial nature of electric dipole moments is taken
into account when the overall moment of a substituted mole
cule is considered: with each group can be associated a
directed dipole moment of a magnitude which to a good ap
proximation is reasonably independent of the nature of the
rest of the molecule. The resultant of the vectorial addition of
the dipole moments for all such groups in the molecule yields
the overall dipole moment. In this way it is easy to appreciate
that the dipole moments of chlorobenzene and the o, m, and
BOX 5: Dipolemoment formulae
Energy of a dipole ju in an electric field E
£ = — (U.E = — p£ cos 8.
Electric field at a point R due to dipole at origin
E (*fc)£ B ** lU
R is a unit vector along R: R= R/fl.
Energy of interaction of two dipoles {general case)
R is the vector from p., to jU^ .
Energy of interaction of two parallel dipoles
3cos 2 0)
(^j/?** 11
R makes an angle with Pi (andpj).
Mean dipole moment of freely rotating polar molecule in a
field E (Langevin function)
<ti) = p£ (iJE/kTi
X U) = coth x — £> tne Langevin function
WpfiEmT) kT>liE.
Molar orien ta tion p olariza tion (LangevinDebye equation)
p is the density, a the polarizabiltty, e the relative
permittivity, M is the molecular weight.
pdichtorobenzenes are 1.70 D, 2.25 D, 1.48 D, and zero
respectively (Fig. D1}.
The computation of electric dipote moments is a surpris
ingly complicated business, and its success depends critically
on the accuracy of the molecular wavefunctions. The difficulty
can be appreciated by considering the simplest possible
approach to their calculation. In this the electron charge
density on each atom in the molecule, plus the positive nuclear
charge of the atom, may be represented as a point charge at
each nucleus. It is then simply a matter of computing the
dipole moment
51
FJG. D1. Vectorial addition of dipole moments illustrated for 1, 2
dichlorobenzene. The experimental moment is 2'25 D.
+qe/2 ^^ ^^+qe/2
cr X)
FIG, D2. Contributions to the dipole moment of H^O: (a) point
charge, (b) asymmetry dipole, (el atomic (hybridization} dipole.
dipole moment of an array of point charges, distributed
according to the molecular geometry. This point charge model
(Fig. D2a) gives bad results because it is a poor approximation
to assume that all the electronic charge is located at an array of
points. One factor omitted is the asymmetry dipole. This arises
from the charge distortion due to the "overlap of orbitals of
unequal size. Consider the case when an occupied orbital is
formed from the overlap of a large porbital on one atom and
a small sorbital on another. The region of maximum overlap
lies closer to the small atom, and so instead of it being reason
able to regard the charge as residing on one atomic nucleus or
the other, there is a considerable accumulation of charge in
the overlap region close to the smaller atom (Fig. D2b): the
asymmetry dipole moment is the contribution to the overall
moment of this extra charge distribution. It is by no means
negligible, for it may amount to about 1 D. Another major
contribution is the atomic dipole, which is also called the
hybridization dipole and which arises, as the latter name
suggests, when an electron occupies a 'hybridized atomic
orbital. When this is so the centroid of electronic charge on an
atom no longer necessarily coincides with the centre of nuclear
charge {Fig. D2c). This separation of charge centroids is
especially important when the hybridized orbital is a "lone
pair; and the atomic asymmetry it is responsible for is
reflected in a contribution to the total dipole moment of the
order of 1 D. The contributions to the total moment of the
water molecule are illustrated in Fig. D2.
Methods of measuring molecular dipole moments include
the Stark effect in molecular spectroscopy and the measure
ment of the dielectric constant (electric relative permittivity)
of solutions (see °polarizability}.
Questions. 1, What does a nonvanishing dipole moment rep
resent tn a molecule? Why is the debye a convenient unit to
use? Why does a molecule with a centre of symmetry possess
no dipole moment? From Table 6 estimate the dipole moments
for o, m, andpdibromobenzene. What are the deficiencies
of the point charge model? What is the asymmetry dipole?
What is its direction? What is the atomic dipole?
2. The energy of a dipole in an electric field E is determined
by — )U*E. Caiculate the energy (in J and in J mol 1 ) required
to reverse the orientation of a dipole of magnitude
52
Dirac equation
1 D{33 X 1CT 30 C m) in a field of 100 V m"*. Which orien
tation has the lowest energy? Consider a dipole moment on a
freely rotating molecule; set up the Boltzmann distribution for
the orientation of the molecule in an electric field, and show
that the mean dipole moment does not vanish, (In fact, show
that the mean dipole moment is given by the Langevin
function {Box 5).) Show that at low applied fields (how low?)
the mean dipole moment is inversely proportional to £7". Cal
culate the degree of ordering, and the energy of the optimum
configuration, when a polar molecule is in the vicinity of an
ion. Choose several separations for the species, which may be
taken to be water and the sodium ion.
Further information. See Coulson (1961), and Chapter 10 of
McGlynn, Vanquickenborne, Kinoshita, and Carroll (1972) for
a thorough discussion and examples of calculations, and also
§8.10. See also Streitweiser (1961) for accounts of the
computation of dipole moments for organic molecules. Tables
of moments, their determination and use 3re given by Smyth
(1955), Smith (1955), Sutton (1955), McCleilan (1963),
and Minkin, Osipov, and Zhdanov (1970). See §15.18 of
Moore (1972), and Gasser and Richards' Entropy and energy
levels (OCS 19) for a simple deduction of the Langevin
expression. Wheatley (1968) gives a simple introduction to the
determination and application of dipole moments. The energy
of interaction between two dipoles, the mean orientation
polarization (the dipole moment of a collection of freely
rotating molecules in the presence of an electric field, as
determined by the Langevin function), and related information
are given in Box 5. See °polarizability for related concepts.
Dirac equation. The Dirac equation describes the behaviour
of an electron in a way that combines the requirements of
quantum mechanics with the requirements of relativity. The
trouble with the Schrodinger equation is that it is unsym
metrica) in space and time (it contains first derivatives with
respect to time and second derivatives with respect to space).
One way out would be to find an equation which had second
order time derivatives; but the unfortunate quality of the
solutions of such an equation (which was proposed by
Schrodinger and named after Klein and Gordon) is that the
total probability of the particle being anywhere in the
universe is predicted to be a function of time; and so the
equation allows the number of particles in the universe to vary.
At the time this seemed unacceptable, and Dirac repaired it by
keeping the equation firstorder in time, and, like Procrustes,
forcing the space derivatives to be first derivatives too. He
could not do this arbitrarily, and the conditions he was forced
to impose led him to conclude that he had to deal with wave
functions with four components. Two components corre
sponded to an embarrassing negative energy, but, instead of
forgetting the whole thing, Dirac had sufficient confidence to
propose that all the negative energy states were filled up
throughout the universe and that we experience only the extra
particles added to the overlying positive energy solutions. How
satisfying it was then when it was shown experimentatly to be
possible to excite the particles out of their negative energy
states, leaving a hole. Such a hole will have a positive charge
and positive mass, and is referred to as the positron (e + ). It
was then shown by Dirac that the two positive energy sol
utions had the same energy, but that when a magnetic field
was applied this degeneracy was removed, one level rose and
the other dropped. This can be interpreted in terms of the
electron having a magnetic moment that can take two orien
tations; in other words it is a particle with 'spin ^. The
magnitude of the splitting was characteristic of a charged
spin^ particle, but with a moment twice as large as would
be anticipated on the basis of a classical model of a spinning
charge. Thus the anomalous °svatue of the electron emerges as
a natural consequence of the Dirac equation.
Purifier information. Dirac (1958), of course. A simple
account of the construction of the Dirac equation is given by
Moss (1973) and by Bjorken and Drell (1964); see also
Landau and Lifshitz (1958a), Schiff (1968), Messiah (1961),
and Schweber (1961) for accounts of increasing sophistication.
See the Dirac Festschrift, edited by Salem and Wigner (1972),
for recent developments.
dispersion forces. The induceddipole— induceddipole con
tribution to the van der Waals intermolecular force is the
London dispersion force. It is a kind of self generating boot
strap force, and arises by virtue of the correlation of fluctu
ations in the electrondensity distribution of neighbouring
dispersion forces
53
molecules. The fluctuation in the electron density on molecule
A causes it to possess instantaneously a dipole moment (if it is
a polar molecule the fluctuation makes an extra ephemeral
contribution to the permanent dipole). This instantaneous
dipole polarizes the electrons on a neighbouring molecule; the
accumulation of electrons on molecule A drives the electrons
from the neighbouring region of B and so induces there, as in
Fig, D3, a region of positive charge. Likewise, the region of
relative positive charge on A matches the region of negative
charge it induces in B. Such a correlation of charge dis
tributions lowers the energy and corresponds to a binding
force. Note that it is essential that there be a correlation
between the charge fluctuations, for otherwise the effect
would disappear.
driving for response)
.fluctuation
response (or driving)
fluctuation
FIG. D3. The induceddipole— induceddipole (dispersion) interaction.
The dispersion energy has a characteristic ft" 6 dependence
{which is reflected in the LennardJones 6, 12°intermolecular
potential} whose source is easy to detect in the model: the
field arising from the initial dipole on A has an ft 3 depen
dence {see Box 6), and the return interaction, where the dipole
on B interacts with the dipole already on A, contributes
another ft" 3 . The interaction energy also depends upon the
"polarizability of the molecules, for it is a measure both of
the ease with which an electric field may distort the molecule
and of the extent of its charge fluctuations. The energy of
interaction of two spherical molecules separated by a distance
ft and with polarizabilities a and a and ionization potentials
/ and / B is given by the London dispersion formula, quoted
in Box 6. Its magnitude is of the order of a few kJ mol~ l . The
ionization potential enters the formula because the extent of
BOX 6: Dispersion energy between spherical molecules
Energy of interaction of a and b separated by ft:
X [d{a) 0n .d(a) mo ][d(b) 0n .d(bU].
where A mn = (E m  E g ) a + (jF ffl  E) b the excitation
energies of a and b,
d 0m (a) = S&npS < a > d ^ m (a), the transition dipole of a,
don <b> = /d7^S(b)d^ (b), the transition dipole of b.
London formula
where a is the polarizability of the atom and / is its
"ionization potential.
If a is in cubic aVigstroms, ft in aVigstroms, and / in elec
tronvolts.
"•WWW
3/327T 2  00095.
Retardation. When ftA/hc > 1
fluctuation depends on the ease with which a molecule can be
excited.
At great distances the dispersion formula fails on account
of the time it takes the information that a fluctuation has
occurred to travel between the atoms: this is the retardation
effect. When ft is large (ft > fie/A, A a typical molecular
excitation energy) the ft 6 dependence is replaced by ft" 7 , and
the interaction falls off more rapidly that the London formula
predicts.
Questions. What are the dispersion forces, and how do they
arise? What is the dependence on the separation of the mole
cules? Why does the interaction energy depend on the
polarizability of the molecules? On what other properties does
54
doubling
it depend? What does the distancedependence become at large
distances, and what is the change due to? Why are hydro
carbons volatile? Why does l 2 dissolve readily in liquid
ammonia? Why is it difficult to liquefy helium?
Further information. See MQM Chapter 1 1 for the deduction
of the quantummechanical expression for the dispersion
interactions, and of the London formula. See also Kauzmann
(1957), Chapter 18 of Ey ring, Walter, and Kimball (1944),
Chu (1967),  Ml of the book of Hirschf elder, Curtiss, and
Bird (1954). A brief survey has been given by LonguetHiggins
(1965); see also Rowlinson (1969) and Curtiss (1967). The
subject of intermoiecular forces is reviewed under that
heading, and more references will be found there.
doubling. In a spectrum 'doubling' refers to a situation in
which the degeneracy of a level is removed by some hitherto
neglected interaction; each twofold degenerate level is split
into two distinct nondegenerate levels— hence 'doubling'. The
doubling shows up in the spectrum either as an actual doubling
of the number of lines, when other states can make transitions
L
U '
V
V \
b
i
FIG. D4. Different types of doubling in a spectrum: (a) the originally
degenerate level combines with one level; (b) the originally degenerate
level combines with two different levels. Unperturbed levels are in
black, perturbed levels are in colour.
FIG. D5. Adoubling: the mechanism in a diatomic molecule.
to the now separated levels (Fig. D4a), or there occurs a situ
ation in which some states combine with (which means make
transitions to) only one of the components of the doublet, and
others combine with the other component (Fig, D4b).
An example of doubling in atomic spectroscopy is the •fine
structure in the spectra of the aikali metals. Take sodium as an
example, and consider an excited configuration in which the
valence electron occupies a 3porbital, then drops down into
the 3sorbital, and in the process emits the characteristic
yellow sodium light (589 nm). Close inspection of the
spectrum shows that the yellow light is slightly impure, for it
consists of two components, one at 588.99 nm and the other
at 589.59 nm; this splitting can be ascribed to a doubling of
the excitedstate levels. More details will be found under fine
structure; but, briefly, the excited state is 2 P, this has two
'levels 2 Pi/2 and 2 P3/2, which are degenerate in the absence of
spinorbit coupling but differ in energy when the coupling is
taken into account. Transitions from these two levels to the
ground state 2 S I/2 yield the two spectral lines separated by
172 cm" 1 .
An example of doubling in molecular spectroscopy is
/^doubling (lambda doubling) in which the rotation of a
linear molecule removes the degeneracy of the two components
of a II "term (or, less importantly, of a A or a <& term). It is
possible to identify the interaction responsible by considering
a diatomic molecule in which a single valence electron occupies
a Jrorbital. In Fig, D5 we illustrate the situation in which the
electron occupies either the orbital tt or the orbital if , where
doubling
55
the superscript is based on the behaviour of the sign of the
orbitals when they are reflected in the plane of molecular
rotation. If the electrons were able to follow the rotation of
the nuclei exactly, with no lag or slip, the two TTorbitals
would have precisely the same energy; this is wellknown to be
the case when a static molecule is considered, as in the usual
discussion of bonding theory. But the electrons do not follow
the motion of the nuclei exactly (see "electron slip and the
"BomOppenheimer approximation), and there is a tendency
for the electrons to lag behind the nuclei. In the case of if the
lag has virtually no effect, for to the lowest approximation the
nuclei slide a little forward in the orbital's nodal plane and so
remain surrounded by roughly the same electron density as in
the static case. But the nuclei slide from the vertical nodal
plane of 7T + into a region of nonzero electron density, and so
it is entirely plausible that the energy of ?r + is modified by the
rotation. It follows that the two orbitals move apart in energy
and that the energy separation, or doubling, depends on the
rate of rotation.
The effect of Adoubling on a spectrum may be seen by
considering the "rotational structure of an "electronic trans
ition in which a diatomic molecule changes from a S state to a
Ft state. (For the technical, we are considering the transition
l II«— '2 + .) The Adoubling of the upper state is illustrated
in Fig. D6: the separation of the components of the state
increases with "rotational quantum number J according to the
ru\eqJ{J + 1 ), where q is a constant. Transitions of the P and
R "branches of the spectrum are allowed to take place from
the rotational states of the lower electronic state to only one
of the components (the one labelled +) of the upper electronic
state, but the Qbranch transitions may take place only to the
other component. From an analysis of the shift of the Q
branch from the centre of the P and R branches it is possible
to extract the value of the parameter q (see Questions) and to
relate this value to the separation of electronic states in the
molecule.
Two other examples of doubling are important. The first is
iltype doubling, in which the degeneracy of a bending vi
brational level of a linear triatomic molecule is removed by
the effects of molecular rotation (see "Coriolis interaction).
The second is 'inversion doubling, where the abi I ity of a mole
cule to invert from one conformation to another doubles its
vibrational spectrum. Qrdoub/ing (omega doubling} is the
analogue of Adoubling when Hund's case (c) is the appro
priate coupling scheme.
Questions. 1. What is the meaning of the term 'doubling'?
How does its presence show in the spectrum of an atom or
J
4
* <
« 4
T
E2q
2 +
1
TT i +
4
I
3
2
1
z* °
p
rR
1
1
1 !
! i
!
L>
Q 1
1 I
FIG. D6. Adoubling: selection rules and spectra. A large Adoubling
of the upper state has been selected. Compare Fig. B12, which is based
on the same parameters.
56
duality
molecule? Discuss the doubling that .arises in the spectra of the
alkali metals: what is the interaction responsible for the
doubling; how does the doubling change as the atom is
changed from Na to Cs; are the "selection rules satisfactory for
the explanation? If the two lines are at 588.99 nm and 589.59
nm, what is their separation in cm" 1 ? Deduce a value of the
spinorbit coupling parameter of Na from these data. What is
the origin of Adoubling? Why should the doubling increase
with rotational quantum number? By inspection of Fig. D5
deduce that the orbital mixed into the 7T + orbital by the
effect of rotation is a fJorbital. Is the orbital mixed in a
bonding or an antibonding orbital? What mixing is expected in
the case of an "antibonding 7rorbital? (It should be stressed
that the actual calculation of Adoubling effects should be
done on the basis of overall states rather than oneelectron
orbltals.)
2. The Adoubling interaction varies as qJ{J+ 1 ); show that in
the spectrum of a diatomic molecule the R and P branches
correspond to a diatomic molecuie with a modified rotation
constant B +b, where B is the true rotational constant, and the
Qbranch corresponds to a molecule with a different rotational
constant B+b . Find expressions for b and b' in terms of q
and show how the displacement of the branches may be used
for the determination of the true B. Construct the first few
lines of the rotational structure of the electronic transition in
which a '£* molecule changes to 'II; in the lower state take
B = 6020 cm" 1 , in the upper B = 6000 cm" 1 , and in the
upper q = 0008 cm 1 .
Further information. See MQM Chapter 10 for a more detailed
discussion, Whiff en (1972), and Barrow {1962). A detailed
discussion of Adoubling with worked examples is given in
Herzberg (1950): see especially pp. 252—5; the basic theory is
laid down on p. 226. Sugden and Kenney (1965) discuss
ikype doubling in § 3.1c, and King {1964) gives a helpful
discussion of Adouhling in § 6.7, fitype doubling in § 9.15,
and inversion doubling in § 9.17. The detailed algebra of
Atype doubling is described by Kovacs (1969). See also
Townes and Schawl ow (1955),
duality. Classical physics dealt with waves and with particles,
and it was quite clear which was which. Quantum mechanics
introduced the view that matter and radiation have a dual
character and that either aspect of behaviour may be exhibited
by the same entity. Thus what for historical reasons had been
classified as particles showed behaviour that hitherto had been
characteristic of waves: it was discovered, by Davisson and
Germer, that crystal lattices could diffract beams of electrons,
and Thomson showed that the same effect could be brought
about by passing the beam through a thin gold foil. Conversely,
it was discovered that what had been classified for historical
reasons as waves showed behaviour that hitherto had been
characteristic of particles: both the "Compton effect and the
•photoelectric effect require the energy or momentum of the
radiation to move around in localized bundles. The conclusion
to which one is forced is that 'waves' and 'particles' have been
so classified because at the level of the experiments done until
the beginning of the present century one type of behaviour
dominated and the other was concealed. Looking more closely
at each type of entity reveals the duality of their character,
and so the nature of matter and radiation is neither just one
nor the other, but a composition of the two. Which aspect
dominates depends on the experiment; no experiment can
exhibit both aspects of the duality simultaneously (principle
of complementarity).
Further information. See Jammer { 1 966) for the way that the
idea of duality emerged as a fundamental property of matter,
and Bohm (1951) for an illuminating discussion. The quantum
mechanical basis of duality may be investigated by dealing
with wave packets and studying the role of the "uncertainty
principle in limiting our view of Nature. We have mentioned
the duality of matter and radiation; further aspects will be
found under °excitons, phonons, "photons, and "polarons.
E
eigenf unctions, etc. When some mathematical operation
(such as multiplication, division, or differentiation) is done on
a function the result is generally soms different function. Thus
differentiation of the function x 2 yields the different function
2x. But some combinations of operations and functions are
such that when the operation is done the same function is
regenerated, but perhaps multiplied by a number. Thus
differentiation of the function exp 2x gives 2exp 2x, which is
the same function multiplied by the number 2. When this
occurs the function is referred to an an eigenfunction of the
operator (in this case the differential operator d/dx), and the
numerical factor (2 in the example) is called the eigenvalue of
the operator. Eigen is the German word meaning 'own' or
'particular'.
The Importance of these names in quantum mechanics can
be recognized by noting that the Schrddinger equation may be
written as Hip = E^/, where H is a differential operator {the
"hamiltonian) and \p is the °wavefunction. This has the form
of an eigenvalue equation, with the energy E playing the part
of the eigenvalue and the wavefunetion as the eigenfunction.
The wavefunction represents a state of the system, and so \p
is often termed the eigenstate, or, because it also shares some
properties with vectors, the eigenvector. The result of an
experiment done on a system in a given eigenstate is the
eigenvalue of the "operator corresponding to the observation.
Questions. 1. Define the terms eigenfunction, eigenvalue,
eigenstate, eigenvector, and eigenvalue equation. Why are these
terms important in quantum mechanics? If you knew the state
of a system, and knew the mathematical form of the operation
corresponding to the observable you wanted to determine,
how could you predict the result of an experiment?
2. Which of the following functions are eigenfunctions of the
operation d/dx : ax, ax 2 , exp ax, exp ax 2 , In ax, sin ax? Which
are eigenfunctions of d 2 /dx 3 ? The operator corresponding to
the component of angular momentum about thezaxis is
(h/i) {3/3 0): if you knew that the state of the system is
described by the function f[r, 0)exp im0, what experimental
result could you predict? What if the state were f[r, 6*)cos m0?
Here is a tricky but important question: what is the eigen
function of the operation 'multiplication by x'f
Further information. See MQM Chapter 4 for a discussion of
operators and eigenvalues. For a deep view of eigenstates and
an explanation of why they are called eigenvectors see Dirac
(1958), von Neumann (1955), and Jauch (1968). See Davydov
(1965) for a helpful summary and examples, including {on
p. 28) the answer to the final question,
Einstein A and B coefficients, "ihe rate of an absorptive
transition between two states is proportional to the density of
radiation present at the frequency of the transition, and the
coefficient of proportionality is the Einstein coefficient of
stimulated absorption B. The rate of emission depends on two
factors. One contribution is proportional to the energy density
at the transition frequency, and the coefficient of proportion
ality is termed Einstein's coefficient of stimulated emission.
Since it is equal to B it is also denoted B. Another component
of the transition rate is independent of the amount of radi
ation already present and is termed Einstein's coefficient of
spontaneous emission and denoted A.
57
58
electric dipole transition
The necessity for this extra coefficient A stems from the
equality of the rates of upward and downward stimulated
transitions: if these stimulated processes were the only ones to
occur the distribition of populations between the upper and
lower states would be equal at equilibrium, whereas the
Boltzmann distribution requires the upper population to be
less than the lower. The spontaneous emission process comes
to the rescue because it provides a way of permitting the
upper states to leak into the lower at a rate independent of
the radiation already present. Thus if the equilibrium popu
lation of the upper state is fJ + , and of the lower is n_, then
the rate of upward, absorptive, transitions is/7 _Bp(U), where
p(v) is the radiation density at the frequency of the transition
of energy/)!', and the rate of the downward transitions is
n t Bp{v) + n t A. Equating the two rates at equilibrium, and
insisting with Boltzmann that n in_ = exp {—hv/kT), enables us
to deduce that the equilibrium radiation density must be of the
formp(p) = {A/B} [exp {hv/kT)— 1]~' . Simply looking at this
expression shows that ,4 cannot be zero if the equilibrium
radiation density is to be nonzero, and so the spontaneous
process is essential. But one can go even further, and find an
expression for A in terms of B by recognizing that at equilib
rium the radiation density will be that of a °blaekbody
radiator, and therefore that p should be given by the Planck
distribution. By comparing the expression above with that in
Box 2 on p. 23 it is easy to deduce that A = 87tf){f/c) 3 S.
An expression for B may be obtained from "timedependent
perturbation theory, and for electric dipole transitions one
finds that B = d 2 /6e h 2 , where d is the magnitude of the
electric transition dipole between the two states of interest.
An important conclusion emerges from the expression for
the ratio A/B, for we see that it increases as the cube of the
frequency of the transition; therefore the spontaneous
processes become very important at high frequencies, but can
be ignored at low frequencies. A consequence of this is the
inherent difficulty of constructing highfrequency "lasers, for
these rely on the cooperative effect of stimulated transitions;
if the spontaneous processes are too important the excited
states decay too quickly and do not contribute to the co
operative laser process.
Questions. 1. What is the significance of the Einstein B
coefficient? What is the rate of transition from a ground
state occupied by N atoms? What is the rate of transition
from an upper state occupied by N" atoms? If A were absent,
show that the equilibrium populations of the upper and lower
levels are equal and in conflict with the Boltzmann requirement.
Deduce an expression for the Einstein A coefficient on the basis
of a Boltzmann distribution of populations, equal rates of
transition at equilibrium, and the Planck distribution of
radiation at equilibrium (p. 23). At what frequencies are the
spontaneous emission processes important? Confirm that the
dimensions of B are those of (volume X frequency) per
(energy X time), and that since the dimensions of p are energy
per (frequency range X volume) the transition rate comes out
with the correct dimensions. The dimensions of A should be
those of time" 1 ; check that this is so.
2. From the Planck distribution (Box 2, p. 23} estimate the
radiation density of light of wavelength 1 cm (microwave
radiation) and 500 nm (visible light) emitted by a biack body
at 1000 K; deduce a value of 8 that corresponds to unit
'oscillator strength and compute the relative importance of
the emissive and absorptive processes at equilibrium.
Further information. See MQM Chapter 7. A discussion of the
A and B coefficients is given in § 12.8 of Davidson (1962),
in § 8d of Eyring, Walter and Kimball (1944), in § 7.2
of Hameka (1965), in Chapter 5 of Heitler (1954), and,
implicitly, by Dirac (1958). Their importance in the discussion
of laser radiation processes is reflected in books on laser
technology; therefore see "laser and references therein.
electric dipole transition. The most intense transitions of
atoms and molecules are those caused by the interaction of
the electric component of an electromagnetic field with the
electric dipole moment of the system. A simple example is the
interaction of a permanent dipole moment of some molecule
with the light beam; when this happens the rotational motion
of the molecule is accelerated and it makes a transition
between two "rotational energy levels. Conversely, a rotating
electric dipole moment behaves like an oscillating electric
charge, and an electromagnetic wave is generated and trans
mitted, by the rotating system. The wave carries away energy,
and so the molecule drops down the ladder of rotational
energy states. The same kind of electric dipole transitions
occur in vibrating systems if accompanying the vibration there
electric dipole transition
59
is a net motion of charge. Thus electric dipoJe transitions can
be induced in the "vibrational energy levels of the polar HCI
molecule because an observer on the side of the molecule will
see a changing charge distribution as it vibrates. The same
observer viewing the chlorine molecule (Clj ( will detect no
such change, and so that. mode of the molecule cannot interact
with the electromagnetic field by an electric dipole mechanism;
it can neither accept energy by absorption nor generate energy
in the field and emit.
Electric dipole transitions may also occur between the
electronic states of atoms, and once again it is necessary to
assess whether the transition involves a motion of charge
which can be interpreted as an oscillation of a dipole. It should
be clear that a transition from a spherically symmetrical 1s
orbital to a spherically symmetrical 2sorbital involves a sym
metrical redistribition of charge, and so no net dipole moment
can be associated with the transition. Therefore this transition
cannot occur via an electric dipolemoment mechanism. If
however we envisage an stop transition, then a dipole
can be identified. If the transition is from s to p the oscil
lating dipole lies along thezaxis, and so the emitted radiation
is transmitted planepofarized with its electric vector in the
zdirection (the intensity would appear greatest to a viewer
stationed in the xyplane). If the transition is from s to p
(or p ) the polarization lies in the direction of x (or y), and
the radiation emerges in a belt largely in the yz (orzx)plane.
When transitions are made between orbitals with definite
values of the 'quantum number m^ the radiation is circularly
polarized; when m^ decreases by unity the emitted radiation is
left circularly polarized, and when it increases by unity the
radiation is right circularly polarized; when mj is unchanged
the light is plane polarized. (We use the convention that the
electric vector of left circularly polarized light rotates anti
clockwise to the observer towards whom the light is travelling.)
This behaviour can be understood by considering the trans
itions s— p; p^ and p may be expressed as superpositions of
states of different mg, and so the radiation emitted is the
appropriate superposition of the plane polarized radiation
already described.
The "selection rules for electric dipole radiation may be
understood in terms of the above discussion and also by virtue
of the "photon's possession of an intrinsic "spin angular
momentum. The first view leads to the Laporte selection rule
which states that the parity of an orbital (see "gerade and
ungerade) must change in an electric dipole transition. Thus in
an s— p transition an even orbital (even with respect to
inversion through the atomic nucleus) changes to an orbital
odd under inversion. The same is true of d— p transitions. The
rule forbids s— s, p— p, and d— d transitions, for with these
there is associated no transition dipole moment. The existence
of a photon angular momentum coupled with the principle of
conservation of angular momentum provides an alternative
view of the situation, because the labels s, p, and d imply that
electrons in the orbitals possess different "angular momenta,
and when a photon is emitted or absorbed the angular
momentum of the atom must change in order to conserve the
total momentum. A photon impinging on an sorbital must
turn it into a porbital, and is unable to turn it into a dorbital
because it brings insufficient angular momentum. When an
incident photon interacts with a porbital the final state can be
either a dorbital or an sorbital, depending on the relative
orientation of the angular momenta in the collision.
The connexion between the change in the value of mn and
the polarization of the light can also be understood on the
basis of the photon spin, because left circularly polarized light
corresponds to one orientation of the photon spin with respect
to its propagation direction and right circularly polarized light
corresponds to the opposite projection. Therefore in order to
conserve the total amount of angular momentum of the total
system about a particular direction, nta must change appro
priately if a photon is absorbed or emitted. If the light
approaches the atom along the negative zax is (from z = — °°)
and is left circularly polarized (spin projection +1 on the
propagation direction, moving like a righthanded screw)
when it is absorbed the atom must change from a state mg to
rr)g+1 if the angular momentum about thezaxis is to be
conserved. The same argument applies to emission.
If an optical transition is observed when the preceeding
rules forbid it, the reason may be that the atom lacks a centre
of symmetry by virtue of its environment, or there may be a
coupling between the electronic and vibrational modes: see
vibronic transitions. Or it may be that a "magnetic dipole or
electric quadrupole transition is responsible for the intensity.
To a very good approximation the electric field of the light
60
electron affinity
cannot interact directly with the spin of an electron, because
that is an internal mode of motion, and so spin is conserved in
electric dipole transitions; when that rule fails it is by virtue
of the spinorbit coupling interaction.
Questions. 1. What are the most common and intense optical
transitions due to? Why is an electromagnetic field able to
accelerate a rotating molecule? To do so what must the mole
cule possess? Can an electric dipole transition increase the
amount of angular momentum about the figure axis (the sym
metry axis) of a symmetric top molecule (a cylindrical or disc
like molecule)? Is the light polarized in rotational electric
dipole transitions? Under what circumstances can a vibrational
mode emit or absorb light by electric dipole transitions? Which
modes of vibration of carbon dioxide are able to interact with
an electromagnetic field, and which are not? (It might be help
ful to revise "normal modes and "vibrational spectra.) Why is it
possible to regard the transitions between states of atoms as
involving an electric dipole moment? What is the name given
to the moment involved? How is the polarization of the light
involved in a transition related to the orientation^ the
electric dipole? Discuss the polarization of the transitions
sp ; , sp x , s~{p x + \p y ), where by (p^ +ip ) is meant the
porbital with m% = + 1. Consider the effect of a magnetic
field on an atom, as in the °Zeeman effect: discuss the polar
ization of the light that you would see as your detector,
sensitive to the three transitions s— p s— p +[ , s— p t , was
moved over the whole sphere of possible orientations. State
the Laporte selection rule: how can it be justified and under
what circumstances will it fail? What role does the spin of a
photon play in determining which electric dipole transitions
are allowed? How may such a description be brought into line
with the Laporte selection rule? Discuss the polarization of
the transitions in the preceeding questions in terms of the
angular momentum of the photon and the conservation of
total angular momenta. Why may the angular momentum of
an atom either increase or decrease when a photon is absorbed
or emitted? If electric dipole transitions forbid a transition,
what might account For its intensity?
2. Discuss the sense in which an electric transition dipole
moment may be said to oscillate at the frequency of the
radiation it transmits (the 'correspondence principle requires
some such sense to exist). Consider the timedependence of
the transition dipole moment by expressing the matrix
element fdr^d^ , where ty is the excited state, ^ is the
ground state of the system, and d is the electric dipole moment
operator, in terms of the explicit timedependence of the
states; that is, write ^ = \jj exp(— if f/h). Apply group
theory {see "character) to the matrix element to deduce the
Laporte selection rule.
Further information. See MQM Chapter 8 for a discussion of
electric dipole transitions. For applications in organic chemis
try see Murrell (1971) and Sandorfy {1964), and for appli
cations in inorganic chemistry, especially in the electronic
spectra of transitionmetal compounds see Orgel (1960),
Ballhausen (1962), and Griffith (1964). The basis of the
selection rules is the calculation of transition probabilities by
timedependent perturbation theory: therefore see MQM
Chapter 7, Herzberg (1940), and Eyring, Walter, and Kimball
(1940).
electron affinity. The electron affinity of an isolated
atom is the energy evolved when an electron is brought up
from infinity and the anion formed. Therefore the electron
affinity is the difference in energy of the neutral atom and
its negative ion: E (X) = £(X)— £(X~). A positive electron
affinity implies that the anion is more stable than the neutral
atom. Since an amount of work E (X) must be done on the
a
ion X" in order to regain the neutral atom and the infinitely
separated electron it should be clear that E (X) is the same as
the "ionization potential /(X~) of the anion: E (X) = /(X").
The determination of electron affinities is often some
what devious. A direct method is to measure the ionization
potential of the anion. The other methods are:
{1 ) electron impact, in which one tries to identify the
appearance potential of the negative ion in the reaction
XY + e > X" + Y;
(2) electron attachment measurements (socalled electron
affinity spectroscopy);
(3) polarography;
(4) application of the BomHaber cycle for the lattice
energy of an ionic crystal.
One application of electronaffinity values is to the construe
electron slip
61
tion of the Mulliken 'electronegativity scale, and it is of use
wherever the stability of an anionic species is required.
Further information. A review of methods for determining
electron affinities has been written by McDowell (1969).
Further information and tables of affinities will be found in
Prichard (1953), Vedeneyev, Gurvich, Kondrat'yev,
Medaredev, and Frankevtch (1966), and Briegleb (1964). For
applications in chemistry, see Puddephatt's The periodic table
of the elements {OCS 3) and Phillips and Williams (1965).
electron slip. Electrons slip not because they are slippery but
because in a rotating molecule they may be unable to follow
the rapid motion of the nuclei: electron slip is a manifestation
of the breakdown of the "BornOppenheimer approximation.
Think of a rotating hydrogen molecule: the electrons may lag
behind the motion of the nuclei as the nuclear framework
rotates. That is, the electrons and the nuclear framework have
slightly different angular momenta.
There are two important consequences of electron slip. The
first is that the distinction between O and U orbital s (and
between S and Fl states) becomes blurred: this is manifest in
the electronic spectra of linear molecules as Atype "doubling.
The other consequence is that all molecules possess a magnetic
moment by virtue of their rotation: this is the molecular
magnetic moment. This moment can be traced to the different
rates of rotation of the positivelycharged nuclear framework
and the negativelycharged electron cloud of the molecule.
There is a net current, and therefore a net magnetic moment.
All molecules possess such a moment when they rotate, and the
magnitude of the moment increases with the "rotational
quantum number J,
Further information. See MQM Chapter 10 for a more detailed
account of the way that electron slip mixes 1> and II states,
and doubling for a picture; for a quantitative treatment see
King (1962), Herzberg (1950), and Kovac's (1969). For mole
cular magnetic moments see "yvalue, §11.6 of Townes and
Schawl ow (1955) and p. 299 of Herzberg (1950).
electron spin resonance: a synopsis. The electron spin
resonance (e.s.r.) experiment is the observation of the energy
required to reverse the direction of an electron spin in the
presence of a magnetic field. The electron possesses a
"magnetic moment by virtue of its spin, and in the presence
of an applied field the two permitted orientations (or and 0)
have different energies (in a fietd of 3 kG theccstate lies
03 cm" 1 above thep*st3te). An electron can be induced to
reverse its orientation (make a transition from 8 to a) if
electromagnetic radiation of the appropriate frequency is
applied, and in a 3 kG magnetic field the 03 cm" 1 radiation
(wavelength 3 cm, frequency 9 GHz) lies in the microwave
region of the spectrum. The apparatus therefore consists of a
magnet capable of providing a homogeneous field in the
vicinity of 3 kG, a source of 3 cm microwaves (a klystron),
and a device for detecting whether the incident radiation is
absorbed. The e.s.r. experiment is performed by maintaining a
constant microwave frequency and sweeping the applied field
until the incident radiation is absorbed (Fig. El ): at this field
the separation between the a and jS orientations exactly
matches (is in "resonance with) the radiation frequency. The
FIG. El, The electron spin resonance transition, and the resulting
spectrum.
electron spin resonance: a synopsis
sample, which must be paramagnetic, may be a solid, a liquid,
or (more rarely) a gas.
Three principal pieces of information emerge from the
experiment:
1 . The position of the spectrum. The magnetic field experi
enced by the electron might differ from the applied field
because the latter is able to induce local fields. For a given
microwave frequency the resonance condition will be attained
at the same local field, and therefore at slightly different
applied fields in different species. If the microwave frequency
is V, so that each photon carries the energy hv, and the applied
field is B, the resonance condition is gfi 8 = hv, where g is a
factor (which, by lack of inspiration, is called the ^factor)
which takes into account the possibility that the local field is
not exactly equal to B. Measuring the position of the spectrum
feld
00 60
FIG. E2. The source of hyperfine splitting in a radical containing one
spin% nucleus (denoted by the larger circle}.
FIG. E3. A typical electron spin resonance spectrum of a radical (with
one spin1 nucleus) trapped in a crystal, Two orientations of the crystal
are shown: note that the centre of the spectrum shifts (9 anisotropic}
and the splittings change (anisotropic hyperfine interactions).
enables g to be determined, and as g depends on the electronic
structure of the paramagnetic species some deductions may be
made about it. In organic and inorganic radicals g is most use
ful for the identification of the species; wider variations in its
value are found in transitionmetal ions, and there it can give
useful structural information, particularly about the separation
of energy levels and the spread of electrons on to ligands. g
may be anisotropic, in which case the position of the spectrum
depends on the orientation of the paramagnetic species {when
it is trapped in a crystal, and the crystal rotated, the resonance
position changes).
2. The hyperfine structure of the spectrum. Generally the
spectrum does not consist of a single line: the structure ob
served is due to the "hyperfine interaction of the electron and
any magnetic nuclei present. A magnetic nucleus (such as a
proton) gives rise to a local magnetic field, which, depending
on the relative orientation of the nuclear spin and the applied
field, can increase or decrease the local fieid experienced by
electron spin resonance: a synopsis
63
the electron spin. This implies that radicals with different
nuclear spin orientations resonate at different applied magnetic
fields, and the spectrum from a large collection of radicals con
sists of lines at all these applied fields (Fig. E2). For example.
N radicals each containing one proton will constitute a sample
consisting of jN radicals with proton spins aligned along the
magnetic field and ~N proton spins aligned against it. The
electrons in the first set of radicals experience one local field
and resonate at the appropriate applied frequency, and the
electrons In the other set resonate at another applied field. The
spectrum therefore consists of two lines separated by several
gauss: this 'hyperfine splitting {h.f .s,} may be interpreted in
terms of the probability that the unpaired electron will be
found in the vicinity of the magnetic nucleus in question, and
so a study of the h.f.s. enables the electron distribution to be
mapped over the molecule. The h.f.s. has both isotropic and
anisotropic components (see Fig, E3): the former is due to the
"Fermi contact interaction and is characteristic of sorbital
character of the electron, the latter is due to the dipoledipole
interaction and is characteristic of porbital character. Therefore
$
1
f
JO
proton spin:
• a
o(3
FIG. E4. A typical solution electron spin resonance spectrum (of
benzene  ), and its interpretation.
hyperfine
interaction
FIG. E5. The spinpolarization mechanism in CH,
a study of the angular dependence of the h.f.s. yields infor
mation on the "hybridization of the unpaired electron's orbital,
and this can be used to discuss the shape of the radical {the
methyl radical, for instance, is shown to be planar). In fluid
solution only the isotropic h.f.s. is observed (see, for example,
Fig. E4), and a principal application is its use to map spin
density in organic radicals. In aromatic radicals the h.f.s. is due
to a spinpolarization mechanism along a C— H bond, as illus
trated in Fig. E5. The jrelectron (with, we choose, Ospin)
causes the aspin in the C— H abond to be predominantly in
its vicinity (electrons prefer to be parallel in atoms: see the
Hund rules) so that the j3spin predominates in the vicinity of
the proton. Therefore, although in the bond the spins are
paired, the proton sees predominantly one spin, and there is a
net interaction.
3. The shape of the spectral tines. The shape of the lines
depends on the type of motion that the radicals undergo, for
it is determined by relaxation processes. Two types of relax
ation process may be distinguished: the spinlattice and the
spinspin. The former arises by virtue of the motion of the
radical giving rise to fluctuating magnetic fields at the unpaired
electron; if these fluctuations happen to have a component
that oscillates at the transition frequency, then a transition can
be induced. The lifetime of the upper state is shortened, and,
by what passes for the "uncertainty principle, the energy of
the state is blurred. This spinlattice relaxation process is
feeble when the motion of the jitterbugging molecule is slow
(because there are no oscillations in the fluctuating field with
64 electron spin resonance: a synopsis
1
anisotropics dominant
width
anisotropics
' overaqed
out
mobility of radicals
FIG. E6. The dependence of the relaxation time on the rapidity of
molecular motion.
the correct frequency 10'° s" 1 ), passes through a maximum
when a typical time scale for molecular motion is 1Q~'°s, and
then declines in extremely mobile liquids when many of the
fluctuations occur at very high frequencies (Fig. E6), The
lifetime of the spin state is called 7"i , the spinlattice relaxation
time, or the longitudinal relaxation time. The longer 7"] the
narrower the line.
The other relaxation process is the spin spin process: in a
fluid but viscous solution each radical is in a particular mag
netic environment or at a particular orientation, and so the
local fields are all slightly different; consequently the radicals
resonate at slightly different applied fields, and the spectrum
is a collection of broadened lines. As the motion of the mole
cules increases the differences in the magnetic environments,
or the anisotropic interactions, are averaged out, and the lines
narrow. The effect of this broadening process is characterized
by the spin spin relaxation time, or the transverse relaxation
time, T 2 : the longer T 2 , the narrower the line. The broadening
effect disappears as the radical mobility increases (by raising
the temperature or reducing the viscosity).
Note that only T x reduces the number of radicals in the
upper spin state, and so only it is a true energy relaxation
mechanism (the direction of energy flow, which is out of the
spin system and into the lattice, or environment, is determined
by the thermodynamics of the system: the entropy increases
for heat flow out of the small spin system into the virtually
infinite lattice). The T 2 process is a different kind of relax
ation process: it relaxes the relative phase of the precessing
electron spins (Fig. E7).
A study of the line widths and saturation behaviour (the
reduction of intensity at high microwave powers as a result of
the equal ization of the a and jS populations) of the spectrum
gives information about the motion of molecules in liquids,
because details of the rapidity of the motion can be inferred
from the shape of the lines. Information can also be obtained
about the rate of chemical processes, for example tautomer
ism, because these also modulate the environment of the
electron spins.
Further information. See Magnetic resonance by McLauchlan
(OCS 1 ) for a simple account of the principles and applications.
FIG. E7. The relaxation processes characterized bv T\ and T%.
electronegativity
65
See also LyndenBell and Harris (1969), Carrington and
McLauchlan (1967), Wertz and Bolton (1972), Slichter (1963),
Abragam and Bleaney (1970) for general accounts. Instrumen
tation is described by Poole (1967). The application of e.s.r. to
organic systems is described in some of these and in Ayscough
(1967); the application to inorganic, nontransition metal
systems is described in Atkins and Symons (1967), the appli
cation to transtttonmetal ions is described by Carrington and
McLachlan (1967), Wertz and Bolton (1972), Slichter (1963),
calculation of the^factor for simple systems is outlined in
MQM Chapter 1 1 , so too is the source of the hyperf ine
structure. See also Memory (1968). Relaxation processes are
described by Carrington and McLachlan (1967), Poole and
Farach (1971), and Standley and Vaughan (1968); and a
collection of important papers has been published by
Manenkov and Orbach (1966). Detailed theoretical exposition
will be found in Muus and Atkins (1972).
electronegativity. The electronegativity of an element in a
molecule is the measure of its power to attract electrons; the
greater its electronegativity the greater its drawing power. Two
definitions of electronegativity are in general use and are due
to the efforts and insight of Pauling and Mulliken.
The Pauling electronegativity scale is based on the lowering
of the calculated energy of a diatomic molecule when ionic
structures are admitted into its description. Suppose the
energy of the molecule AB is truly f(AB) but that a calcu
lation based only on purely covalent structures yielded
E eou (AB), the ionic resonance energy would be E. (AB) —
F(AB) — £" cov (AB). Pauling found that the square root of
f ion (AB) could be set proportional to the difference of two
numbers, one characteristic of the element A and the other of
B, and that the expression [f jon (AB}] K = b£)£l
was valid for a wide range of combinations. In order to set up
the scale it is necessary to estimate E (AB): Pauling proposed
that a reasonable approximation would be the mean of the
energies of the molecules A 2 and B2 . This was justified by the
view that they are manifestly nonpolar and that the energies
of A 2 , AB, and B 2 should form a simple sequence if polar
structures in AB are omitted. Pauling used both an arithmetic
mean and a geometric mean to set up his table (see Table 7),
Having set up the electronegativity scale it is possible to
deduce a number of molecular properties. First, one can use it
to estimate bond energies if the A 2 and B 2 bond energies are
known: this is just a reversal of its definition, but energies
may be predicted of molecules other than those used to
construct the scale. Next one may use it to predict the
tonicity of a bond (the percentage ionic character), and for a
scale expressed in electronvolts the ionicity is given by the
expression 161^^ — X^ ' "+" 3' 5 'X^"'XJ^ 2  From the percent
age ionic character it is possible to give a rough estimate of the
"dipole moment if the bond length is known.
The other scale, that due to Mulliken, is rather more funda
mental, for instead of defining it in terms of unmeasurable
but estimable quantities he defines it as the mean of the
"ionization potential and the electron affinity of the atom,
and both these quantities are measurable. Thus the Mulliken
scale is based on x^ 1 = \ t/{A) + EJA)] . A further advantage
of this scheme is that it is possible to take into account the
differences in electronegativities of different orbitals on the
same atom or the dependence of the electronegativity on its
state of 'hybridization.
It is not surprising that the two scales are related: at a
numerical level it is possible to equate X^ ~ X^ 1 with
2 78 (x^ ~ X^)' at a deeper level it is possible to show that to a
good approximation the value of E. (AB) is determined by
the energy required to move an electron from A to B, which is
E (B) — /(A), and the energy to move it from B to A, which is
E (A) — /(B), and that as in a nonpolar molecule these are
equal (because neither tendency wins) on rearranging the
equality in this case of vanishing electronegativity difference
we find that EJA) + /{A) is equal to E (B) + /(B). Conse
quently the difference between these quantities should be
proportional to the difference of the electronegativities of the
two atoms.
Questions. What does the electronegativity of an element
measure? Why is it a plausible approximation that E (AB) is
the mean of £(A 2 ) and E(B 2 )? Given that the bond energies of
the halogens X 2 are 21 7 eV for F a , 2475 eV for Cl 2 , and
1971 eV for Br 2 , that the energy of H 2 is 4476 eV, and that
of 2 is 5*080 eV, estimate the bond energies of the various
heteronuclear diatomic molecules that may be formed.
66 electronic spectra of molecules: a synopsis
Estimate the ionic character of each molecule. Deduce an
expression for the dipole moment of a diatomic combination
in terms of the electronegativities of its components. Why
should you expect poor agreement with experimental values
(even if you derive a decent expression)? What is the Mulliken
definition of electronegativity? In what sense is it superior to
the Pauling scale? On what grounds would you expect the
Pauling and the Mulliken scales to be related? What molecular
energy terms are ignored in this argument?
Further information. See §5.8 of Coulson (1961) and § 2.1 1
of Pauling (1960) for a detailed discussion of the role and
deduction of electronegativities. For Mulliken's analysis see
Mulliken (1934) and Moffitt (1949b).
electronic spectra of molecules: a synopsis. A natural
progression into complexity is from the electronic spectra of
diatomic molecules into polyatomic molecules, then to the
consideration of chromophores in complex molecules, and
then to the spectra of molecules in solids.
The electronic spectrum of a diatomic molecule contains a
number of bands, each resulting from a transition from the
ground electronic state (X) into an excited electronic state
(A, B, . . .), and the structure of the bands is due to the
simultaneous excitation of vibration; the intensity of these
"progressions is determined by the "FranckCondon principle
and the other "selection rules. On the vibrational structure
there is a further structure due to the excitation of rotation,
and P and R "branches are generally visible, and Qbranches
in some cases. Since the moment of inertia is different in the
two electronic states the branches tend to a head {see
Fig. B12on p. 31): when the moment of inertia is larger in the
upper state than in the lower the Rbranch, the branch to high
frequency, has the head. The vibrational lines get closer to high
energy because the vibrational levels converge towards the
dissociation limit: the dissociation energy may be determined
by observing the dissociation limit or by a careful extrapol
ation from lower frequencies (a BirgeSponer extrapolation is
often used). In some cases the rotational and vibrational struc
ture disappears and then reappears before the dissociation
limit is reached: this is a manifestation of "predissociation.
From the spectrum may be determined the 'forceconstants,
dissociation energy, and "anharmonicity of the electronic
states of the molecule, and the parameters in the molecular
potentialenergy curves.
The electronic spectrum of a polyatomic molecule is more
complicated, but the same principles apply and analogous
information obtained. The return to the ground state is an
additional subject of interest in polyatomics, especially when
the molecule loses its energy by "fluorescence or "phosphor
escence. In a condensed phase the rotational structure of the
spectrum is lost, and the vibrational spectrum becomes so
diffuse that the absorption spectrum is often just a series of
broad bands. The electronic spectra of transitionmetal com
plexes is of particular importance, and the transitions may
often be associated with the "crystalfield splitting of the
delectrons: see 'crystal fie Id theory and °ligandfteld theory.
In other cases the transition is a charge transfer transition from
the metal ions to the ligands: see "colour. In organic molecules
the absorption can often be associated with the presence of a
chromophore such as the carbonyl group (tt**— n electric dipole
transition) or a double bond {n*< — k transition): see "colour.
When a molecule is a part of a crystal lattice other effects
may be observed; one especially important phenomenon is the
formation of an "exciton, in which the excitation hops
through the lattice.
Further information. See MQM Chapter 10 for a discussion of
molecular electronic spectra. Books that summarize the
applications of electronic spectra to the study of a molecular
structure include Barrow (1962), Whiffen (1972), Dixon
(1965), King (2964), Jaffe and Orchin (1962), Rao (1967),
Stern and Timmons (1970), Herzberg (1950, 1966), Murrell
(1971), andGaydon (1968). Further information about the de
tailed topics will be found in the appropriate sections. Gaydon
(1968) is a very good source of information about how to
interpret electronic spectra of small molecules and how to
extract potentialenergy curves and dissociation energies.
Murrell (1971) is concerned with larger, organic molecules,
and discusses individual chromophores and their interaction in
solids. Herzberg (1950, 1966) is a mine of detailed information,
and is a brilliant example of the application of theoretical con
cepts to the detailed examination of molecular properties.
elect ronvolt
67
electronvolt. The electronvolt (eV) is the energy acquired by
an electron when it is accelerated by a potential difference of
1 V. Since the charge on the electron is —1 602 X 10~ 19 C the
energy is equivalent to 1602 X 10" 19 J or 9649 kJ mol 1 .
Another useful conversion is 1 eV = 8023 cm" 1 .
equ ipartltion theorem. The mean energy of each mode of
motion of a classical system in thermal equilibrium is jnkT,
where n is the number of quadratic terms (of displacement or
momentum) needed to specify its energy. As an example, the
energy of an atom in free space arises from its kinetic energy
which can be expressed as the sum of the three quadratic terms
P 2 „l2m,pjl2m, and pi 12m; and so the mean energy of an atom
in equilibrium at a temperature T is ?kT, and 1 mol of
monatomic gas will have art internal energy j RT. A molecule
that can "rotate around three axes will have a mean energy of
wkT associated with each mode, and so a mean rotational
energy of jx7". A "harmonic oscillator has an energy
lp 2 /2m) + ^kx 2 , where x is the displacement from equilibrium,
and so with each oscillatory mode of a body there will be
associated a mean energy k T because two quadratic terms
appear in the expression for its vibrational energy.
The theorem is a deduction from the Boltzmann dis
tribution for the population of energy levels at thermal equi
librium, and the assumption that the modes concerned are
classical. When the modes of motion are quantized the
theorem fails, and so it is not applicable to the vibration of
molecules nor to small rotating molecules at low temperatures.
Questions, 1. State the equipartition theorem. Calculate the
total mean energy of a diatomic molecule, a linear triatomic
molecule, a bent triatomic molecule, and methane, on the basis
that ihe translational and rotational motion are classical and
the vibrational motion quantized and not excited. Derive an
expression for the molar "heat capacity of these species. In
terms of the equipartition theorem discuss the contribution to
the total energy and the heat capacity of a methyl group in a
molecule which at high temperatures is able to rotate freely
about one axis, and at lower temperatures can execute only
classical torsional motions about the axis.
2. From the Boltzmann distribution at a temperature 7" show
that components in the total energy of the form x 2 and p 2
both contribute an amount ^kT to the mean total energy.
What would happen to the equipartition theorem in the event
of quartic terms (x 4 andp 4 ) being involved in the energy?
What effect has •anharmonieity of the form x and x ?
Further information. This subject is really within the realm of
statistical mechanics; therefore see Gasser and Richards'
Entropy and energy levels (OCS 19) for some of its elementary
applications. See also §7.5 of Reif (1965) for a helpful dis
cussion with applications and §10.3 of Davidson (1962) for a
deduction and discussion. §332 of Fowler and Guggenheim
(1965) and §44 of Landau and Lifshitz (19586) are worth
looking at. See Fowler (1936) and Tolrnan (1938) for
erudite discussion.
equivalent orbital. A molecular orbital is one of a set of
equivalent orbitals if a "symmetry operation applied to the
molecule transforms the orbital into another member of the
set. As an example consider one of the C— H bonds in methane,
CH 4 : this is formed from the overlap of an sp 3  hybrid orbital
on the central carbon and a 1sorbital of the hydrogen. If the
tetrahedral molecule is rotated into another equivalent
position another of the sp 3  1s bonds is rotated into the
original position. The four Obonds constitute a set of equiv
alent orbitals. The concept of equivalent orbitals is closely
related to "hybridization and "localized orbitals.
Further information. See Appendix 9.2 of MQM and Chapter 8
of Coulson (1961). Further information can be traced through
"localized orbitals.
exchange energy. The Coulombic interaction between two
electron distributions is repulsive, and its magnitude may be
calculated by dividing both regions into minute charged
volumes, calculating the Coulomb energy of interaction
between each of the charged volume elements, and then
summing over all the elements to obtain the result which we
shall write./ (see "Coulomb integral). Unfortunately this gives
the wrong answer because we have omitted the effect of "spin
correlation, which can cause electrons of the same spin
orientation to stick together and those of opposite spin to stay
apart. Thus if the two electrons have the same spin the true
average repulsion energy will be less than J, because of the
68
exciton
intrinsic tendency of such electrons to avoid each other. The
correction we should make changes the average repulsion
energy to J—K, the correction term K being the exchange
energy. The name reflects the source of the correction which
lies in the °Pauli principle and the behaviour of wavefunctions
when electrons are exchanged.
Further information. See Chapters 8 and 9 of MQM for a more
detailed account and examples of the application of the
concept in atoms and molecules. Good discussions will be
found in §19.1619 of Bohm (1951} and §10.89 of Davydov
(1965). For applications to molecules see Coulson (1961),
Richards and Horsley (1970), and McGlynn, Vanquickenborne,
Kinoshita, and Carroll (1970), who give, in Appendix E, a
guide to the computation of the exchange integrals.
exciton. Imagine an array of identical molecules in a crystal,
and let one be excited. This excitation hops from molecule to
molecule, and slides throughout the crystal until it decays. The
excitation moves like a particle, and this notion is conveyed by
the word exciton, which is the name given to this migrating
excitation. If we think of the excitation as being caused by the
removal of an electron from one orbital of a molecule (or atom
or ion) and its elevation to a higher orbital, then the excited
state of the molecule can be envisaged as the coexistence of an
electron and a hole. The hopping of this electronhole pair
from molecule to molecule is the migration of the exciton.
When the electron and the hole jump together from molecule
to molecule as they migrate we have the tightbinding case,
and the migrating excitation is a Frenkel exciton. It is also
possible for the electron and hole to be on different molecules,
but in each others vicinity; this is the weakbinding case and
the migrating excitation, now spread over several molecules
(more usually ions), is a Wannier exciton. In molecular solids
the Frenkel exciton is more common, and that will be our
interest.
The formation of an exciton affects the spectrum of a
species in a solid. This should not be surprising, for the mi
gration of an exciton implies that there is an interaction
between the species composing the crystal (for otherwise the
excitation onone unit could not move to another), and this
interaction should affect their energy levels. The strength of
the interaction governs the rate at which an exciton moves
through the crystal: strong interaction implies fast migration;
vanishing interaction implies that the exciton is localized on its
original molecule, in which case it is an ordinary excitation of
an 'isolated' molecule, Exciton formation causes lines to shift,
split, and change in intensity. The reasons for this we shall
understand when the mode of interaction between the mole
cules has been explained.
An electronic transition in a molecule involves a shift of
charge (see "transition probability and "electric dipole tran
sition). A shifting charge on one molecule exerts a force on a
neighbour, which can respond with a shift of its charge. This
process of transition dipole interaction can continue, and the
excitation can rattle through the crystal (Fig. E8).
The process can be looked at in a slightly different way by
considering all possible relative orientations of the transition
dipole moments of the molecules of the crystal, and then
seeing what combination a light wave can excite. Let us
consider a linear array of molecules with transition moments
perpendicular to the line (the arrows indicate the effective
direction of motion of the charge when a molecule is excited).
On the basis of simple electrostatics the energy of the array
with all transition dipoles parallel is higher than any other
phasing (Fig. E9), and so a transition to this arrangement of
dipoles throughout the crystal appears at an energy higher
than the transition to any other arrangement, and indeed
higher than the transition in the separated molecules. But it is
only this arrangement that the incident light is able to excite.
This is because the wavelength of light is so long in comparison
with the molecular spacing that its electric field has the same
phase over a large number of molecules; therefore it stimulates
a whole domain of transition dipoles to move in phase. This
means that the presence of the exciton coupling appears in the
spectrum as a shift of the absorption band to high energy. If
FIG. E8. An excitation migrating through a crystal as an exciton.
exciton
high energy
Ikjhf stimulates rhis mode
7r\ /3\ //
isolated
molecule
low energy
isolated molecule A A exciton
FIG. E9. Transition dipoies with various phasings. The alternating
alignment lies lowest, the parallel (the one excited by incident light)
lies highest.
the transition dipole moments were along the line of the mole
cules we should get the opposite shift; this follows from the
fact that the inphase excitation of the transition dipoies gives
a headtotail array, which has a low energy.
If there is more than one molecule per unit cell it is possible
to obtain several lines: N molecules per unit cell give N absorp
tion lines, or exciton hands. The splitting between the bands is
the Davydov splitting. To see how it arises consider the case
when N = 2 and the molecules are arranged as shown in
Fig. E10. Let the transition dipoies be along the length of the
molecules. The light field stimulates the inphase excitation of
the transition dipoies, but they need be inphase only in so far
as neighbouring unit cells are concerned. Within each unit cell
the transition dipoies may be arrayed as in Fig. E10 a or b, and
these have different interaction energies. This will appear in
the spectrum as two bands split by the energy of interaction
between the transition dipoies within the unit cell.
Questions. 1. What is implied by the term 'exciton'? What is
the difference between a Frenkel and a Wannier exciton?
Would you expect excitons to contribute to the conduction of
electricity in the medium? How may excitons be recognized
spectroscopically? Why is it reasonable to consider only the
inphase excitation of transition dipoies? What will happen to
the spectrum as this approximation fails? Assess the wave
length at which it might be expected to be seriously in error
for the 380 nm band of anthracene.
2, Estimate the interaction energy between two transition
dipole moments separated by 0.3 nm and each of magnitude
1 D (33 X 10" 30 Cm). Suppose the "oscillator strength of the
transition in the free molecule is f = Q2; estimate the exciton
shift. What is the source of the Davydov splitting? What is the
polarization of the split bands?
Further information. A simple account of the formation of
molecular excitons will be found in Murrell (1971), who also
discusses hypochromism (the reduction in intensity of absorp
tion) and hyperchromism (the increase in intensity). (Further
solated molecules
A
^
Davydov splitting
FIG. E10. Davydov splitting for exciton bands for two molecules per
unit cell.
70
expectation value
confusion may be encountered with the names hypsochromism ,
which implies the lightening of a colour, as in dyeing, and its
opposite, bathochromism , the deepening of a colour.) An
introduction to the mathematical theory of excitons will be
found in Craig and Walmsley (1968) and Kittel (1971), and
further development is given by Davydov (1962, 1965).
expectation value. The expectation value of an observable is
its mean value for the state of the system, and is the mean
result of a set of experiments designed to determine the value
of the observable for that state. Thus the expectation value of
the energy is the mean energy that would be measured in a set
of suitable experiments on a collection of identical systems;
the expectation value of the position is the mean position that
would be measured, and the expectation value of momentum
or angular momentum is that mean value.
If the system is in a pure state with respect to one of the
observables (in other words the state is an eigenstate of
the "operator corresponding to that observable) all identical
experiments give identical results, and the expectation value is
the result that would be measured in all the experiments; such
results are dispersion free. An example would be the determin
ation of the energy of an atom or a molecule when it is in a
definite energy state; it is then in an eigenstate of the energy
operator for the system, and the expectation value of this
operator is the energy that would be measured in all
experiments.
Quite often the system is not in a pure state but is better
described by a superposition of pure states; performing one
experiment to measure a property of such a system gives a
result which can be identified as one of the eigenvalues of the
operator corresponding to the property, and the mean result
of a set of identical results performed on a set of identical
systems will be the expectation value for the system in that
state. An example could be the determination of the linear
momentum of a system which is described by a "superposition
of states of different momentum {for example, if the particle
is described by a wave packet): if the state with momentum
kh occurs in the superposition with weight lc. P then the
experiment will yield the answer kh with a probability \c k I 2 ,
and will yield other values with a corresponding probability.
The average of all such measurements is the expectation
value of the linear momentum, written (p>.
If the state of a system is \p the expectation value <S2>
of an observable £1 is equal to the integral fdT\p*£l\]j , where
£2 is the appropriate operator for the observable, and it is
supposed that the state \p is normalized. Therefore we may
predict the result of a set of experiments by calculating the
expectation value by evaluating the integral. But in one
experiment we can expect to observe the value of the integral
only if the state \p n is an eigenstate of the operator corre
sponding to the experiment we are attempting to perform. If it
is not an eigenstate, the value of the integral tells us only the '
mean value of a large number of experiments; this is the only
information we are able to calculate, and we are unable to
predict, except as a probability, precisely what result we shall
get from a single experiment.
Questions. What is meant by the expression 'expectation value
of an observable'? In which sense does it determine the result
of an experiment? What interpretation should be put on the
expectation value when the state of interest is a mixture (a
superposition)? What is meant by a pure state? A beam of light
is constituted from two planepolarized orthogonal com
ponents and the resulting beam can be described as the super
position c x \p x + Cyipy, where X and Y refer to the two
polarization directions. Discuss the result of an experiment
designed to measure the polarization of the beam. Using the
expressions in Box 15 on p. 166 calculate the expectation value
of the kinetic energy and the linear momentum for a particle
in a onedimensional square well, and discuss the results of an
experiment to confirm the result.
Further information. See MQM Chapter 4 for a discussion of
operators, observations, and expectation values. See also
Feynman, Leighton, and Sands (1963), Kauzmann (1951),
Pauling and Wilson (1935), and Bohm (1951). For deeper
discussions of the measurement process see Dirac (1958),
von Neumann (1955), and Jauch (1968).
extinction coefficient. The intensity of a beam of light is
diminished as it passes through an absorbing medium, and
since the amount each molecule absorbs is proportional to the
extinction coefficient
71
intensity of the light present, the intensity falls exponentially.
The rate of decay of the intensity is determined by the extinc
tion coefficient. The actual intensity follows the BeerLambert
law I = j exp(—cell), where / is the initial intensity, c is the
molar concentration of the absorbing species, £ the path length
(which by convention is normally expressed in centimetres),
and e is the extinction coefficient. The extinction coefficient
depends on the frequency of the incident light, and is often
written e[v), where V is the corresponding wave number in
cm 1 .
The extinction coefficient is a measure of the "transition
probability at the appropriate frequency, and is therefore a
measure of the strength of the "transition dipole and the
•oscillator strength of a transition, and in turn is related to the
"Einstein coefficient of stimulated absorption. Molar extinc
tion coefficients for a few representative materials are
recorded in Table 8. The product ce{i>)$. is called the optical
density of the material at that frequency.
Further information. See MQM Chapter 10, Appendix 10.2
for the relation of extinction coefficient and oscillator
strength; for this relation see oscillator strength. A discussion
of the Beer Lambert law and its applications and limitations
will be found in Wayne (1970). A compilation of extinction
coefficient data and a discussion of its analytical applications
are given by Mellon (1950).
F
Fermi contact interaction. A magnetic nucleus and an
electron may have a magnetic interaction by virtue of their
contact: this magnetic interaction is the Fermi contact inter
action, and it is a special case of a "hyperf ine interaction.
But what is the nature of the interaction on 'contact'? It is
possible to give a variety of explanations, of varying sophisti
cation, of the actual mode of interaction; the most pictorial is
as follows. Consider the magnetic moment of the nucleus as
arising from the circulation of a current: we replace the
magnetic moment by an equivalent current loop. At distances
far from the nucleus the field due to this loop is indistinguish
able from the field from a point magnetic dipote; but close to
the nucleus, or loop, the pointsource nature of the field is
invalid and the magnetic field is characteristic of a circular
FIG. Ft. An interpretation of the Fermi contact interaction: only an
electron in an sorbital can penetrate the nucleus.
loop of nonvanishing diameter (Fig. F1). Now bring in the
electron with its spin "magnetic moment. Far from the nucleus
it experiences a pure dipolar magnetic field, but if it can pen
etrate the nucleus it enters a quite different region, where the
field flows in only one direction. The magnetic interaction
between this nondipolar field and the electron's magnetic
moment is the contact interaction.
The electron must come into contact with the nucleus if
the interaction is to operate. Can an electron come into con
tact with the nucleus? If it is a pelectron, or a delecton, etc.
it cannot, because all such orbital s have a mode at the nucleus.
An sorbital has no node at the nucleus, and an electron occu
pying one has a nonvanishing probability of being at the
nucleus. It follows that only selectrons can show a Fermi
contact interaction. Since sorbitals are spherically sym
metrical it also follows that the interaction should show no
directional characteristics, and indeed it is found to be
isotropic.
How strong is the interaction? A measure of its strength is
the extra magnetic field that an selectron experiences by
virtue of its interaction with the nucleus. For the Isorbital in
the hydrogen atom the contact interaction (of energy 1420
MHz) is equivalent to a magnetic field of 508 G acting on the
electron; for an electron in a 2sorbital of fluorine the field is
as strong as 17 kG. The inner electrons of heavier elements can
possess tremendously large interactions, amounting to
megagauss. Some representative values are listed in Table 9.
The Fermi contact interaction plays an important role in
^electron spin resonance spectra, because the extra local field
appears in the spectrum as hyperfine structure. The contact
72
fermion
73
interaction, being isotropic, does not vanish in fluid media. It
also plays an important role in "nuclear magnetic resonance
because it is a contribution to one of the mechanisms of
"spinspin coupling.
Questions. 1, What is the Fermi contact interaction? In what
sense does it depend on the non vanishing si2e of the nucleus?
Why is the contact interaction characteristic of an selectron?
Why is the interaction isotropic? Where does the interaction
play an important role?
2. The energy of the Fermi contact interaction is given by the
expression — {2fi Q /3) [y J e l^(0)l 2 ] 1. 1, where 7 N and y e are
the rnagnetogyric ratios of the nucleus and electron respect
ively (see "magnetic moment), I and s their spins, U the
vacuum permeability, and li^O)! 2 the probability that the
electron is at the nucleus. Show that this interaction has the
form suggestive of a magnetic field (Uq/S) y N \\p{0}\ 2 I arising
from the nucleus and affecting the electron magnetic moment,
and from the expression for the hydrogen atomic orbitals given
in Table 15 on p. 275 compute the magnetic field (actually the
magnetic induction) experienced by an electron in a Is and
2sorbital of hydrogen. Repeat the calculation for a selection
of °Slatertype atomic orbitals for the firstrow elements. In
each case replace I by its maximum projection m h.
3. An electron in an orbital centred on a neighbouring nucleus
may have a Fermi contact interaction whatever the nature of
the orbital, provided the orbital has a nonvanishing amplitude
at the nucleus of interest (thus a 2porbital on B may have a
nonzero amplitude at the nucleus of its neighbour A). The
strength of the interaction is determined by the expression in
Question 2 but with f^(0)P replaced by l^(R)l 2 , where R is
the position of the nucleus relative to the nucleus on which
the orbital is centred. Calculate the strength of the interaction
for a proton as it is brought towards the position of another
proton surrounded by a 1 selectron. Ignore the distortion that
the second proton induces.
Further information. See MOM Chapter 11 for a further
discussion of the interaction. An account of the role the inter
action plays will be found in McLauchlan's Magnetic resonance
(OCS 1) and in Atkins and Symons (1967), Carrington and
McLachlan (1967), Slichter (1963), and Abragam and Bleaney
(1970). Magnitudes of the interaction for a variety of nuclei
are given on p. 21 in Atkins and Symons (1967). Calculations
involving the contact interaction are described by Memory
(1968) and Freeman and Frankel (1967). The derivation of
the contact interaction is described in a simple manner in
MQM Chapter 1 1, and performed in §4.5 of Slichter (1963); a
derivation from the Dirac equation is given in Griffith (1964)
and Bethe and Salpetre (1957).
fermiOfl. A fermion is a particle possessing an intrinsic "spin
angular momentum characterized by a half integral spin
quantum number (s or /). Examples include the electron
(s = h, the proton (/ = j), the neutron (/ = j), the neutrino
(s = ), 3S C1 nucleus (/=§), 13 C{l = j), and 17 (/ = §).
Ferrnions obey the Paul! exclusion principle, and so no more
than one can occupy any single quantum state. This has a
profound influence on their behaviour, and distinguishes them
sharply from bosons, any number of which may enter a given
state. The "Pauli principle requires a wavefunction to be
anti symmetrica I under the interchange of any pair of identical
ferrnions.
Further information. See spin and the °Pauli principle for
further discussion. The occupation restriction of ferrnions is
taken into account by FermiDirac statistics when large
collections are under consideration: these are described by
Gasser and Richards in Entropy and energy levels (OCS 19), in
Chapter 6 of Davidson (1962), and in §9.7 of Reif (1965).
fine Structure. The fine structure in an atomic spectrum is
the splitting between different levels (different values of J of a
particular "term). In atomic sodium the energy of 2 Pj/2 differs
from the energy of 2 P 3 /2, and so the emissions 2 Pi /7 — * 2 Si/2
and z P 3/2 — ► z Si /2 occur at slightly different frequencies (and
give rise to the two closelyspaced yellow D lines of the sodium
spectrum); this is an appearance of fine structure.
Fine structure is a manifestation of spinorbit coupling and
is best introduced by considering a oneelectron atom. Suppose
that the electron in the atom has an orbital angular momentum
1: by virtue of its charge it also has a "magnetic moment,
which by Ampere's law may be considered to be a dipole at
the centre of the orbit. This magnetic moment gives rise to a
magnetic field which interacts with the spin magnetic moment
74
f i nestructu re co nsta nt
of the electron, and the interaction energy depends on the
relative orientation of the electron's spin and orbital magnetic
moments. This implies that the energy also depends on the
relative orientations of the angular momenta 1 and s (see
Fig. F2). The lowenergy orientation corresponds to the
opposition of ji. and /i„, and therefore it also corresponds to
the opposition of 1 and s; consequently the lower value of the
total angular momentum j (j = 1 + s) corresponds to the lower
energy.
FIG. F2. The magnetic interaction resulting in fine structure. A high
total angutar momentum corresponds to a parallel alignment of
moments, and therefore to a high energy.
We see that different values of/ (and of J in manyelectron
atoms) correspond to different energies by virtue of the
magnetic spinorbit interaction. This is the basis of the third
°Hund rule: when an electron shell is less than half full, low
values of J have lower energy than high values of J (the
opposite is true when the shell is more than half full). "Terms
in which high/ levels have higher energies than low/ levels
are called regular; when low/ levels lie highest the term is
inverted. Since a term with several/ values (for example, a 2 P
term which has the two levels 2 P[/2 and 2 P 3 /2) is called a
"multiplet term, we arrive at the names regular multiplet and
inverted multiplet.
Within a given multiplet the spacing of the levels obeys the
Lande interval rule which states that the energy interval be
tween pairs of adjacent levels is in the ratio of the /values of
the upper level of each pair. Thus the ratio of the 3 Pj— Pj
interval to the 3 P,  3 P interval is 2:1.
As the spinorbit coupling increases with atomic number we
both expect and observe the fine structure to be very import
ant in heavy atoms.
Further information. The fine structure is discussed in more
detail in MQM Chapter 8. See especially §111 A. 5 of Kuhn
(1962), where the mathematics is developed in a simple way;
atoms more complex than hydrogen conform to the same
principles, and are discussed by the same author in §lll D.3
and Chapter V. See especially Chapter 2 of Herzberg (1940).
Discussions of the multiplet structure and fine structure of
atoms is also described in detail by Woodgate (1970), Condon
and Shortley (1963), Griffith (1964), and Shore and Menzel
(1968).
fineStructure constant. The finestructure constant is a
measure of the strength of interaction between a charged
particle and the electromagnetic field: it is given by the
expression a = e~/4Tre hc; it is dimensionless and has the
numerical value 1/13703602, or approximately 1/137. The
smallness of this number is of great importance, for it
determines the size of atoms and the stability of matter. If
<x were much larger the distinction between matter and radi
ation would be much less clear; if it were much smaller, matter
would have virtually no electromagnetic interactions.
These considerations can be elucidated by considering how
the finestructure constant (that is, the strength of the coupling
between charged matter and the electromagnetic field) deter
mines the size of atoms and the magnitude of some of their
properties. Consider an atom of radius r in which the potential
energy of the electron is of the order of Ze 2 /47re r: it is tempt
ing to express this in terms of a, and falling into temptation we
obtain Zcihclr. The kinetic energy of the atom is of the order
of p 2 /2m ; if we use the uncertainty principle to assess the
order of magnitude of this term on the basis of an electron
being confined to a region of radius r we may use p~hlr. This
enables the total energy to be expressed in terms of the
fluorescence
75
parameter r, roughly the size of the atom, and to seek r by
minimizing the total energy (finding the value of r for which
df/dr = 0). The answer we get is rX /Za, where X is the
"Compton wavelength of the electron (A_ = h/cm ~
24 X 10 m). This shows that the size of an atom is roughly
this characteristic 'size' of an electron times 137. It is
interesting to note that the "Bohr radius can be expressed as
X_/47Ta, or as r Id 2 , where r is the classical radius of the
electron. If the interaction strength were much less the atom
would be much larger, and if a were much larger, and com
parable to the analogous coupling constant for the nucleon
interaction (the 'strong interaction' as opposed to the electro
magnetic interaction), then atoms wouid be of roughly the
same size as their nuclei. The minimum energy can be found
by substituting the size of the atom back into the energy
expression, and rather pleasingly we i'md  IrZ 2 a 1 (m c 2 ): this
is pleasing because m c is the relativistic expression for the
rest energy of a particle of mass m , and so the energy of an
atom is of the order ofZ 2 /137 7 of this value. This is a small
proportion when Z is small, but it may approach unity as Z
gets large (in heavy atoms). This implies that ordinary non
relativistic quantum mechanics is good for light atoms, but
fails progressively through the periodic table. Another amusing
deduction is the order of magnitude of the velocity of an
electron in an atom or molecule: simple consideration of the
energy expression (see Questions) gives the answer v~Zac:
since a is so small, electrons move at nonrelativistic velocities
except in the case of heavy atoms. Another place where the
finestructure constant enters is in the magnitude of the "spin
orbit coupling which determines the "fine structure of spectra:
in hydrogenlike atoms the strength of this interaction is pro
portional toa 2 ff, where R is the Rydberg constant, and it is
from this relation that a gets its name.
The stability of atoms is determined at several levels by the
small size of a. One important role played is the way that the
"transition probability for electric dipole radiation (essentially
a mechanism whereby an atom emits or absorbs a photon)
depends on a: an analysis of the situation shows that an elec
tron has to oscillate about 1/a(aZ) 2 times before it is virtually
certain to emit a photon: this accounts for the moderate
stability of the excited states of atoms and molecules. The
probability that two photons are thrown off by an excited
atom is of the order of the square of the probability that one
will be emitted, and so this process is of very low probability
and is indeed only rarely observed. But matter can also decay
into radiation: matter can be annihilated and appear as
electromagnetic energy. The probability of an electron dis
appearing in this fashion is proportional to its strength of
coupling to the electromagnetic fields, and varies as a. Put
another way, an electron spends a of its time as electro
magnetic radiation. Fortunately a is small; if it were closer to
unity, matter and radiation would be indistinguishable.
Questions. 1. Of what is the finestructure constant a measure?
What would be the consequences of it being zero? Evaluate
its magnitude from the values of the fundamental constants,
and confirm that it is dimension less. What features of atomic
and molecular structure does it determine? Express the spin
orbit coupling parameter, the °Bohr radius, the Rydberg
constant, and the "Einstein coefficients in terms of a, and
attempt to interpret the form of the expressions. Carry through
the calculation of the hydrogenatom energy as described in
the text: find the lowest energy as a function of atomic size r,
the size of the atom in this case, and the velocity of the elec
tron (from the kinetic energy).
Further information. A very good and quite simple account
of the way that the finestructure constant determines the
magnitude of atomic properties is given in Chapter 1 of
Thirring (1958).
fluorescence. Light when absorbed by a molecule may be
degraded into thermal motion or it may be reirradiated. Light
emitted from an excited molecule is fluorescence or phos
phorescence. The two types of radiation may be distinguished
by the mechanism that generates them and, more ambiguously,
by the general observation that fluorescence ceases as soon as
the exciting source is removed, whereas phosphorescence may
persist. See "phosphorescence for an account of its mechanism.
The mechanism of fluorescence is illustrated in Fig. F3. The
incident light excites a groundstate molecule into a state which
we shall label S t ; assuming the ground state is a singlet state
(all electrons are normally paired in molecules) and we are
interested in a strong initial absorption, the upper state is also
a singlet. The molecule is also excited vibrational ly during the
FIG. F3. The processes leading to fluorescence, and the 'mirror* image'
relation of the absorption and Fluorescence spectra. Note the shift of
the latter to longer wavelengths.
transition (see FranckCondon principle). Collisions with the
surrounding medium, which may be a gas, a solvent, or a solid
lattice, induce vibrational transitions because the surrounding
molecules may be able to carry away the moderately small
amounts of vibrational energy of the molecule, and so to lower
it down the ladder of vibrational states in the upper electronic
state. When the molecule has reached its lowest vibrational
state two things may occur. One is that the solvent may be
able to carry away the electronic energy and so deactivate the
molecule: this the solvent may do if it has an energy level that
matches the energy of the excited molecule, for there may
then be a 'resonant transfer of radiation to the solvent, which
then fritters away its excitation into thermal motion. An
alternative mode of decay, and the one that concerns us o1
present, is the fluorescent decay of the excited electronic state:
the molecule deactivates by emitting a photon and falling back
into the lower electronic state. This emitted light is the
fluorescence.
Whether or not fluorescence appears depends on compe
tition between the radiattonless deactivation, involving energy
transfer to the surrounding medium, and the radiative emission.
In a gas a molecule receives about 10 10 collisions per second,
and in a liquid the continuous jostling by the solvent amounts
to about 10 12 — 10 13 collisions per second. Observed lifetimes
of fluorescent radiation are of the order of 10~ 8 s, and so in a
gas we have to consider the effectiveness of about 100
collisions, and in a solvent 10 4 — 10 s collisions. If the
collisions are strong in the sense that they are effective in
absorbing energy then the molecule will be deactivated by the
radiationless processes, especially when the molecules are in a
liquid. If the collisions are unable to extract much energy,
then even in a solvent they may be able to lower the elec
tronically excited molecule down only the vibrational ladder,
and they will be unable to extract an electronic excitation
energy. In this case the radiation decay dominates and the
molecule fluoresces.
Two characteristics of the fluorescence should be noted.
The first is that the fluorescence should appear at lower
frequency than the incident light. This may be seen im
mediately from Fig. F3, which shows that the energy of the
emitted photon differs from that absorbed by the amount of
vibrational energy lost to the surrounding medium. Therefore
we might expect objects irradiated with blue or ultraviolet
light to fluoresce more in the red. Brightred fluorescent
clothing is a part of the modern scene, and a manifestation of
this effect. The second point is that there may be vibrational
structure in the spectrum of the fluorescent light: this
vibrational structure is a "progression formed by the decay of
the ground vibrational state of the upper electronic level into
different vibrational levels of the lower electronic level. Its
study can provide information about the "force constant of
the molecule in its ground state, and this is in contrast to
normal electronic spectra which provide information about the
stiffness of the bonds in the upper electronic level. It follows
that the absorption spectrum and the fluorescence spectrum of
a molecule should resemble each other: this is normally ex
pressed by saying that one is the 'mirror image' of the other
{see Fig. F3); but that description should not be taken
literally, because the vibrational splittings and intensities are
not quite the same.
A number of details may be added to this basic description.
The first is that the initial absorption might not take place to
the lowest excited singlet state of the molecule. In this case an
fluorescence
77
internal conversion occurs in which the higher singlets S 2 , S 3 ,
etc. are induced, by a collision, to make a radiation less trans
ition into the lowest excited singtet S] , which then fluoresces
(see °Auger effect). A famous rule due to Kasha reflects this
effect: the fluorescent level is the lowest level of that multi
plicity (for example, the lowest excited singlet level). The
intensity of fluorescence depends strongly on the physical
state of the sample because of the deactivating collisions in
competition with the fluorescence; pure, undiluted liquids
generally have a very low fluorescent efficiency because the
excitation may hop from one molecule to an identical
neighbour by a resonant process (see °exriton for the
analogous effect in solids). Conversely, it is possible to en
hance fluorescence by having present a molecule that can
absorb the incident light and then transfer it (by a matching
of energy levels, and a collision) into a molecule that may then
fluoresce: this is sensitized fluorescence and is made use of in
some kinds of "laser. Another term often encountered is
resonance fluorescence; this signifies that the fluorescent
radiation has the same frequency as the incident light; when
this is so the fluorescence may be brighter because the
transition is stimulated. Light of exactly the same frequency
is rare in fluorescing molecules because the presence of the
solvent slightly shifts energies, and so the (00) vibrational
upwards transition {that is the transition from the lowest
vibrational level of one state to the lowest of the other) might
differ in energy from the (0—0) downwards transition because
the solvent may solvate the upper state differently before the
fluorescence occurs. Fluorescence is generally extinguished as
soon as the incident illumination ceases: this is because al) the
transitions of interest are allowed, and therefore occur very
quickly. Neverthefess, there is the phenomenon of delayed
fluorescence (not to be confused with phosphorescence)
which may persist for several milliseconds. The mechanism
for this depends upon the excitation of a molecule from the
ground state S into the singlet S l( and then this molecule
migrating to another molecule to which it sticks by sharing its
excitation. Thus we have the reaction S + S* — > (SoS^*.
This excimer (if the two molecules are the same) or exciplex
(if they are different) then falls apart, after a short life, with
the emission of fluorescent radiation,
Questions. What are the characteristics of fluorescent radiation?
How may it be distinguished from phosphorescent radiation?
What electronic processes are responsible for fluorescence?
What other alternative paths of energy degradation are open to
atoms and molecules in gases and solutions? What properties of
the solvent determine the rate of nonradiative decay? What
differences would you expect in the fluorescent behaviour of
a molecule dissolved first in a strongly interacting solvent with
highfrequency bending and stretching vibrations, such as water,
and then in a weakly interacting solvent with flabby bonds, for
example selenium oxychloride? What is a fluorescent spec
trum of a molecule? How does this information complement
the absorption spectrum? In what sense is the fluorescent spec
trum a mirrorimage of the absorption spectrum? What is the
role of the °FranckCondon principle in determining the
structure of the fluorescence spectrum? What is meant by the
term 'internal conversion", and what is its significance in the
study of fluorescence? In what sense is fluorescence an "Auger
process? Why is fluorescence largely quenched in the pure
liquid? How may fluorescence be enhanced? What is resonance
fluorescence? In what sense does the operation of some types
of *laser depend on the mechanisms described here? Why does
the fluorescence spectrum show a shift from the position of
the absorption spectrum, even allowing for the mirrorimage
symmetry? To answer this, consider a molecule surrounded by
a polar solvent; and then excited by a transition that changes
the molecule's polarity (for example, a jr*«— n electric dipole
transition). Let the polar solvent relax about the excited state,
and consider the energy of the emissive transition. What is
meant by the terms excimer and exciplex, and what is a conse
quence of their formation? How may delayed fluorescence be
distinguished from "phosphorescence?
Further information. See MQM Chapter 10 for more discussion.
A simple account of some of the relaxation processes described
has been given by Heller (1967) in connexion with liquid lasers.
See also Haught (1968). A good account of fluorescence and
related processes has been given by Bowen (1946), Wayne
( 1 970), and Calvert and Pitts ( 1 966) . The generation of I ight
in chemical reactions {chemiluminescence) is a phenomenon
related to fluorescence, the difference being that the excited
78
force constant
state of the emitting molecule is formed as the product of a
chemical reaction. This subject is described by Wayne (1970).
Energytransfer processes are at the root of the fluorescence
efficiency; therefore see Levine and Bernstein (1974).
forceconstant. The forceconstant k is the constant of
proportionality between the restoring force and the dis
placement x of a simple harmonic oscillator: force = ~kx.
Large force constants imply stiff systems (strong restoring
forces even for small deviations from equilibrium). The
frequency of a classical simple harmonic oscillator is related to
k by the expression o> = {klm) v ' , where m is the mass of the
oscillating system (with k expressed in newtons per metre and
mass in kilograms, the frequency will be in radians per second;
to get hertz divide by 2ir). The lesson taught by this expression
is that the frequency of oscillation is determined by the mass
as weli as the forceconstant, for the heavier the mass the less
effective will be the restoring force. In the quantummechanical
treatment of the "harmonic oscillator the energies are given in
terms of the fundamental frequency loq calculated in the same
way as the classical case: the energy of the nth quantum level
is (n + jJ^^
The importance of the force constant in quantum theory is
that it is a measure of the stiffness of bonds between atoms,
and therefore governs (together with the atomic masses) the
"vibrational frequencies of molecules.
Table 10 lists typical values of forceconstants for some
molecules and the corresponding frequencies and quantum
energylevel separations.
Questions, 1. What is the forceconstant? What is the physical
significance of a large force constant? Would you expect the
forceconstant for the C— C bond in diamond to be less than
that for the Pb— Pb bond in metaliic lead? A mass of I kg
hangs from a spring with forceconstant 1 N m" 1 : what is its
natural frequency in radians per second and in hertz? To what
should the mass be changed in order to oscillate at iHz? What
forceconstant would be needed if the mass were that of a
proton and the frequency that typical of a molecular bond
(~10 l4 Hz)? What wave number (cm 1 ) does this correspond to
in the infrared absorption spectrum? Where is the absorption
shifted to on deuteration?
2. By a Taylor expansion of the bond energy about its
equilibrium value, show that the forceconstant is proportional
to the curvature of the molecular potential energy curve at the
equilibrium separation.
Further information. See MQM Chapter 10 for more infor
mation about molecular vibrations. A good, simple, but
detailed account is given by Woodward (1972), who reveals
how to determine k from vibrational data, and a standard work
is that of Wilson, Decius, and Cross (1955), See also Whiffen
(1972), King (1964), and Gans (1971). A complication of
molecular vibrational data has been made by Adams (1967,
1971).
FranckCondon principle. The FranckCondon principle
governs the intensity of spectral transitions between the
vibrational levels of different electronic states of molecules.
By recognizing the great difference in mass between the
nuclear framework and the electron being excited, it states that
the electronic transition occurs so rapidly that during it the
nuclei are static. A vertical transition occurs, which begins with
the nuclei in some arrangement in the lower electronic state
and ends with them in the same arrangement in the upper
electronic state. But as the molecular energy curves might be
displaced (see Fig. F4) this nuclear arrangement might corre
spond to a highly compressed or stretched state of the excited
molecule, and so the molecule immediately starts to vibrate. It
follows that a vibrational excitation of the molecule generally
accompanies an electronic transition.
Which of the vibrational levels is most populated by the
transition is governed by the relative positions of the upper
and lower energy curves: if the curves were of the same shape
and one lay directly over the other (Fig. F4a), the transition
would be from the ground vibrational level of one to the
ground vibrational level of the other, and so the electronic
transition would occur without vibrational excitation. In
general, we may envisage the transition as occuring from the
most probable conformation of the ground state— which is the
static, equilibrium arrangement of the nuclei. The electronic
transition occurs, and during it the nuclei do not change their
arrangement. At the completion of the electronic transition
the nuclei are static, but in a new forcefield because of the
FranckCondon principle
79
FIG. F4. The classical basis of the
FranckCondon principle. The bob
remains static during the excitation,
and the amount of vibrational
energy simultaneously excited
depends on the relative disposition
of the potentialenergy curves.
new electronic distribution. They therefore begin to move, and
swing harmonically away from and back to their initial
arrangement (Fig, F4b and c). It follows that the original
arrangement is a turning point of the new motion, and that
vibrational energy is stored by the molecule. A line drawn
vertically from the initial ground state intersects the upper
potential energy curve at the point which will be the turning
point in the excited state, and which shows how much energy
is absorbed in the transition. (Remember that the energy of a
"harmonic oscillation is constant: what potential energy it
loses as the spring decompresses is turned into kinetic energy
which is used to recompress the spring. Therefore the potential
energy at the turning point, E in Fig. F4, determines the
energy at all displacements for that mode of oscillation.)
The quantummechanical basis of the principle is the
"overlap between the "vibrational wavefunctions of the two
electronic states: transitions occur most strongly between
vibrational states that overlap most, because two states that
overlap strongly have similar characteristics. The ground
vibrational wavefunction is a bellshaped curve with its
maximum at the equilibrium nuclear conformation (Fig. F5).
Many of the vibrational wavefunctions of the excited elec
tronic states overlap this function, but the greatest overlap
occurs with functions that peak in the same region of space. If
the energy curves are displaced, the peaks of importance are
those that occur at the edge of the potential well (see Fig. F5),
and so the vibration excited will be that predicted by the
simple device of drawing a vertical transition from the equi
librum separation in the ground state (the most probable con
formation, and where the vibrational wavefunction peaks
strongly) to the point where it intersects the edge of the upper
potential curve. In the vicinity of this intersection the vi
brational wavefunctions have moderately strong amplitudes,
and so overlap most strongly with the ground vibrational
state. The observed distribution of vibrational intensities then
reflects the different overlaps between the ground and excited
vibrational state wavefunctions.
Questions. 1. State the FranckCondon principle. On what
does its validity depend? What is the 'classical' explanation of
the principle? What is the quantummechanical explanation,
and how is the classical explanation related to it? What is a
vertical transition? Construct a diagram similar to that in
Fig. F5 in which the FranckCondon principle is applied to
determine the intensity distribution of "fluorescent transitions
(from the ground vibrational level of the upper electronic
state to various levels of the lower electronic state).
2. Demonstrate the quantummechanical basis of the principle
by considering the "transition dipole moment between the
80
Fra nek Hertz experiment
FIG. F5. The quantum basis of the FranckCondon principle. The
strongest transition occurs to the state with which the lowest vi
brational level of the lower state has the greatest overlap; this is
shown shaded. The resulting spectrum is shown on the left.
two states. Proceed by supposing that the state of the mole
cule can be written as the product of the vibrational state and
the electronic state, and then think about the transition
moment <e*, v* Idle, v). Show that this may be approximated
by an expression of the form (e* ldle> {V* If), and recognize
the presence of the overlap integral between the vibrational
levels of the electronic states. Calculate the overlap integral
between the ground states of two "harmonic oscillators whose
equilibrium conformations are displaced by a distance R, and
plot the (00) transition intensity (the transition V = to
V = 0) as a function of R.
Further information. See MQM Chapter 10 for a further
account, and a deduction of the principle. See also §6.16 of
King (1964), Whiffen (1972), and Barrow (1962). A thorough
discussion of the basis of the principle is given in §IV,4 of
Herzberg (1950) and developed in even more detail in
Herzberg (1966). The original formulation of the principle
was by Franck (1925), and this was turned into mathematics
by Condon (1928). A useful summary of modern work in
volving the principle has been given in Nicholls (1969).
FranckHertz experiment. In the FranckHertz experiment
a beam of electrons was passed through a gas at low pressure
and the current arriving was monitored; in later experiments
the energy of the electron arriving was also monitored. As the
energy of the incident electrons was increased it was found
that the current arriving at a collector dipped sharply when the
incident energy was equal to some excitation energy of the
atom or molecule. It was observed that the sample simul
taneously emitted light of a frequency corresponding to the
energy of the incident beam. This can be rationalized in terms
of the quantization of energy, for the electrons are able to
donate their kinetic energy to an atom only if the atom can be
excited by that amount; therefore the current will dip each
time the energy can be imparted to the quantized system. This
is confirmed by the observation of the emitted radiation at the
corresponding frequency [hv= Af).
One of the important features of the experiment is that it
is an illustration that energy is quantized even when it is
imparted from mechanical motion, as opposed to electro
magnetic radiation.
Further information. See Chapter VI of White (1 935) for a useful
introduction. More discursive accounts are given in §2.15 of
Bohm (1951) and p. 85 of Jammer (1966). The original papers
are those of Franck and Hertz (1914, 1916, 1919).
free valence. An atom may be linked to its neighbours by
bonds of various order ( "bond order); the free valence of the
atom is the difference between its maximum possible total
bond order and the actual total bond order, and it therefore
reflects the lack of saturation of the valence requirements of
that atom. If triple bonds are discounted, for carbon the
maximum total bond order is 473 (or 3 + /3). As an example,
in the 'benzene molecule each carbon atom has a bond of
order 1 to its hydrogen atom, two abonds to the neighbouring
carbons, and two ?rbonds of order \; therefore the total bond
free valence
81
order of each carbon is 433 and the free valence is 040, In
butadiene the C— C bond orders are 189, 145, and 189 along
the chain, and so the free valences are 0'84 for the two outer
most atoms and G39 for each of the inner pair.
The magnitude of the free valence is a quantitative measure
of Thiele's early theory of partial valence, which allows some
predictions to be made about the relative reactivity of atoms
in conjugated chains. Thiele supposed that because only one
bond was sufficient to hold two carbon atoms together each
carbon atom in the chain had a partial valence which could
be used for reacting: the remaining part of the double bond
was superfluous. Nevertheless he took the view that the two
central partial valences got tangled up, leaving only the outer
atoms with available valencies. Now, however, we can interpret
the reactivity in terms of the different free valencies on the
atoms. Free valencies also occur at the surfaces of metals, and
the chemical consequences of this include the immensely
important role of catalysts.
Further information. See §9.1 1 of Coulson (1961), and §2.9
and several other sections in Streitweiser (1961). See also
Daudel, Lefebvre, and Moser (1959), and Pullman and
Pullman (1958). For lists of bond orders see Coulson and
Streitweiser (196S). For a discussion of the reactivity of
molecules in terms of the concepts of free valence, and other
quantities, see Chapter 1 1 of Streitweiser (1961). For a dis
cussion of catalytic activity see Bond's Heterogeneous catalysis:
principles and applications (OCS 18). Why 473? See
Moffitt (1949a).
G
gvalue. The "magnetic moment of an electron that arises
from its orbital angular momentum is 7 1, where 7 is the
magnetogyric ratio; but the magnetic moment of an electron
due to its spin angular momentum is 07 s, where g is an
additional 'anomalous' factor to which experiment ascribes
the value 20023 (often approximated to 2). Like most
'anomalous' quantities an explanation can be found in a
deeper theory, and indeed it should not be surprising to find
an extra factor of 2 appearing in connexion with spin (which,
after alt, has no classical analogue, and involves />a/fintegral
quantum numbers). The deeper theory required is that of
"Dirac: his relativistic quantum mechanics leads naturally to
the deduction that g = 2; but the theory is too strict, for it
requires g to be equal to the integer 2. This shows the Dirac
theory to be incomplete: the extra 0/0023 required for
the observed value can be found from the even deeper
theory of "quantum electrodynamics. In this theory the
electron is continuously buffetted by stray electromagnetic
fields which are always present, even in an ideal vacuum: these
fields affect the spin of the electron in such a way, that the
magnetic moment is increased from its Dirac value. The cal
culation of the 0value of the free electron is one of the
triumphs of quantum electrodynamics.
The LandS g factor is closely related to thejvalue we have
just described; indeed, they are identical in the limit of vanish
ing orbital angular momentum. The La nd e jf actor determines
the effective magnetic moment of an electron or atom possess
ing both spin and orbital angular momenta, which are com
bined together to give a total angular momentum J. For an
atom described by the quantum numbers S, L, and J the Lande
tff actor is ffj= 1 + ^ {[JU + 1} + S(S+ 1)£(L+ 1)]/
J{J + 1 )}. When L = we obtain g J = 1 2 because J can then
equal only 5; and when S = we obtain g, = 1, which is the
normal value for a spinless system. In terms of the Lande
factor the magnetic moment of a system with angular
momentum J may be written g,y J.
The peculiar form of gj arises from the anomalous pvalue as
follows. The vectorcoupling picture of a system with spin
momentum S and orbital momentum L coupled to give a
resultant J is shown in Fig. G1. L and S both "precess around
J, and J processes around some other axisz. Antiparallel to
both L and S we may draw vectors representing the corre
sponding magnetic moments, but the spin magnetic moment
must be drawn twice as long in proportion to the orbital
moment because of the factor g = 2. The resultant of ji.
and « s , denoted ji'j, does not lie along the direction of J (it
would if g = 1, as shown in Fig. Gib}, but will precess about
it because of the precession of L and S about J. Only the
component along J does not average to zero during this
precession, and so the effective magnetic moment of J, which
we write fy, depends on this component, and therefore on the
value of L and 5. When L> S or S> L,^j lies almost com
pletely along J and g, is approximately t or 2 respectively,
because either the orbital or the spin moment is dominant; but
when L and 5 are similar, the effective part of ju' may be much
smaller than its true magnitude. Because J processes around the
zdirection so too does ju,; therefore if the projection of J on
z is J z , the projection of ju, on z hg.y J , where g, is some
82
gvalue
83
FIG. G1. In (a) is illustrated the
source of the Lande'flfactor;
[b] is a hypothetical situation in
which g  1.
factor to be determined from the geometry of the situation
depicted in Fig. Gla. It turns out that this factor is the Lande
factor already quoted.
The gvalue in 'electron spin resonance is related to both
these ^values. In this technique, which is described in more
detail in the appropriate section, an oscillating field of fre
quency V is brought into "resonance with a spin system held in
a magnetic field B. The energy of the oscillating field is bv,
and the energy separation of the electron spin levels (that is,
the energy separation between m — + ^ar\6fn = — 5") is equal
foffjUgS: the resonance condition is therefore hU — gii^B. The
gf actor in this expression would be the freeelectron value <7 if
the electrons under consideration were free, or the Lande'ffj
value if they were bound to an atom. In fact the electron investi
gated is normally part of a molecular system, and so neither situ
ation holds. To a very good approximation (except in some
transitionmetal ions) an electron in a molecule possesses no
orbital angular momentum; therefore we can expect the^vaiue
of a radical to be very close to 2 , 0023. Nevertheless, the orbital
angular momentum is not completely "quenched because of
the presence of the spinorbit coupling interactions. Normally
there is sufficient spinorbit coupling to leak some of the spin
angular momentum into orbital angular momentum, and so we
should expect the jvalue to fall slightly below 2 , 0023 towards
the value it would have if the momentum were all orbital and
not spin (that is, towards 1 ). This is widely observed, and
ffvalues of the order 20000 and thereabouts occur frequently.
Furthermore, the deviation from the freespin value increases
with the magnitude of the spinorbit coupling constant for the
radical, and this accords with our interpretation. For a rough
orderofmagnitude estimate one may write the deviation from
20023 as J/A, where f is the spinorbit coupling constant and
A a typical excitation energy of the radical {see "perturbation
theory and "fine structure; some Rvalues are listed in
Table 9). This mechanism can also account for the observations
of yvalues exceeding the freespin value, which is the case
when the electron is a member of a shell more than half full,
but the argument is slightly more involved. Basically it is
connected with the reversal of the order of levels when a shell
is changed from being less than half full to more than half full.
We know that according to the °Hund rules the levels become
inverted (low/ levels lie above high/) because the sign of the
spinorbit coupling changes. This can be interpreted in terms
of changing from a system in which electrons are orbiting to
one in which holes in an otherwise completed shell are
circulating. A hole carries, in effect, a charge opposite to an
84
g value
electron's, and the inversion of the /levels, and the deviation
of g to values above the freespin value can be traced to this.
A final kind of ffvalue is the molecular gvatus , which
relates the rotational angular momentum of an entire
molecule to the magnetic moment that arises from its motion.
Even a closedshell molecule, such as H 2 or methane, possesses
a magnetic moment by virtue of its rotational motion; it arises
because of "electron slip and the consequent imbalance of the
rotating negative and positive charges leading to a net circu
lating current, and thence to a magnetic moment. The
imbalance increases as the rotational motion quickens, and so
the magnetic moment is proportional to J, the rotational
angular momentum. It is normally written /i — gy J, and g,
which depends on the details of the molecular electronic
structure, is the molecular 5value. Mote that it is common to
use the nuclear magneton (/i^ = eh/2m ) to define the
molecular Rvalue: molecular magnetic moments are so small
that this is more appropriate than using the Bohr magneton
(using j± N makes g of the order of unity). For ammonia
g ~ 053, and so the magnitude of fi when the molecule is in a
state with J  10 is approximately 53/i .
Questions, 1 . What electronic property does the gf actor
determine? In an atom it is appropriate to label g with the
value of J: why does the atomic magnetic moment depend on
J, L, and S? Evaluate the Lande gf actor for atoms with S = 0,
L=1;S = j, t = l;S=1,i = 1;S = 0, L = 2: evaluate it for
all possible values of J in each case. The energy for a magnetic
moment with zcomponent fi in a magnetic field 6 in the
zdirection is — fj. B: evaluate the energies of the states of the
atoms for which you have just calculated the Lande' g factors.
Now repeat the calculation with the false assumption that the
^factor for the free electron is 1 and not 2, Draw your results
on a ladder energylevel diagram in order to see how the
coupling of L and S modifies the magnetic properties of a
system. Turn to "Zeeman effect for further development of
this point. What does the^factor measure in electron spin
resonance? Why can g differ from 2 when the electron is a part
of a radical? What information about the electronic structure
of a radical cang reveal? Calculate the magnetic field for
resonant absorption of 9 GHz radiation of (a} a free electron,
(b) a radical with 5 = 20057, and (c) a radical withff= T9980.
Quote results in gauss (G). What is the role of spinorbit
coupling in determing the^value of a molecule? Why can
someffvalues be more than the freespin value? What is the
molecular Rvalue, and how does it arise?
2. This question invites you to deduce the form of the Lande
fffactor. We consider the geometry set out in Fig. Gla with a
magnetic field along z. We seek to express the energy of the
system as  sryy e J.B, whereas in fact we know that the
magnetic energy is really — 7 L.B — 27 S.B (that is, the sum
of the orbital and spin interactions with the field). We also
know that J = L + S. Consider L. This vector precesses about
J, and the only timeindependent, and therefore nonvanishing,
component is the one parallel to J; this has the value L.J/I J I.
This component now behaves like the vector [(L J)/J] (J/UI),
swinging round thezaxis as J itself precesses (J/J is a unit
vector along J). This vector has a projection along the magnetic
field direction which we may write [(L.J)/lJ l] [J.B/U I] , and
so the only timeindependent component of —7 L.B is
— 7 e [LJ/UI 2 ) J.B. This has the form —g/f J.B, which is
what we require. A similar expression for —27 S.B can be
written as — g .7 J.B (find it), and so now we must show that
g, + g, = g,, the Lande factor. Do this by noting that
2L.J = (JL) 2 J 2 L 2 = S 2 J 2 L 2 , and similarly for
S.J, and then finally replace operators of the form J 2 by their
quantum mechanical values J(J + 1)h 2 . Find a more exact
form of the Lande factor using g = 2'0023 instead of a "~ 2.
e e
Further information. An account of the deduction of the
Landedfactor will be found in Chapter 8 of MQM. and a further
discussion is given in §1 1 .3 of Herzberg (1940), and §IIIF
and §VA3 of Kuhn ( 1 969). The factor plays an important role
in the "Zeeman effect and in determining the magnetic
properties of transitionmetal ions. For references to the latter
see "crystalfield theory, and ligandf ield theory. A detailed
account will be found in §5.6 of Griffith (1964) and §11.3
and subsequent chapters of Abragam and Bleaney (1970). For
theffvalue of electron spin resonance see MQM Chapter 1 1 for
a detailed discussion, including something more about <7
exceeding 2*0023, and an account of the calculation of ^values
for molecules. This topic is also taken up in Chapter 2 of
Atkins and Symons (1967), in Chapter 9 of Carrington and
McLachlan (1967), in Chapter 7 of Slichter (1963), and in
gaussian atomic orbitals
85
Chapter 12 of Griffith (1964). McLauchlan describes the
dependence of the appearance of an electron spin resonance spec
trum onthepvalue in Magnetic resonance {OCS 1 ).
Molecular magnetic moments are discussed in Chapter 11,
especially §11.6, of Townes and Schawlow (1955).
gaussian atomic orbitals. The bore about molecular
structure calculations is the complexity of many of the
electronelectron interactionenergy integrals which must be
calculated. In a selfconsistent field calculation many of the
integrals involve atomic orbitals based on more than one centre,
and may involve orbitals on as many as four centres. These
multicentre integrals are complicated to evaluate and consume
a great deal of time on an electronic computer. One simplifi
cation is to express the atomic orbital In terms of gaussian
functions, which are basically of the form exp(— ar 2 ), instead
of as the Slater type of atomic orbital, which are basically
exponential functions of the form exp(aV). The advantage of
this procedure arises from the fact that the product of two
gaussians based on different centres is itself a gaussian based on
a point lying between the centres. Therefore a complicated 3
or 4centre integral can be expressed as a relatively simple 2
centred integral, and this can be evaluated speedily. The dis
advantage of the method lies in the fact that an atomic orbital
is not well represented by a simple gaussian function, and so
each atomic orbital has to be expressed as a sum of several
gaussians. Therefore, although each integral is simpler, very
many more of them need to be evaluated.
Questions. What advantages stem from employing gaussian
atomic orbitals, and what are the disadvantages? Show that the
product of two gaussian functions exp(— a,r\) and exp(— a 3 r)
may be expressed as a gaussian function centred on a point
between the two centres, and find the appropriate position in
terms of a t and a t . Confirm that the procedure cannot be
applied in the case of two exponential functions. By the
variation principle determine the best gaussian atomic orbital
for a hydrogen atom; repeat the calculation using a trial
function composed of the sum of two gaussians. In each case
compare the true 1sorbital with the best gaussian orbital by
comparing the energies, and by plotting the radial dependence
of the functions.
Further information. See §9.2(i) of Richards and Horsley
(1970), and §1.6F of McGlynn.Vanquickenborne, Kinoshita,
and Carroll (1972). The method was originated by Boys
(1950). For calculations based on the method see the bibli
ography prepared by Richards, Hinkley, and Walker (1971 ).
gerade and ungerade {g and u).The German words gerade
(even) and ungerade (odd) are added as labels to states and
wavef unctions to denote their behaviour under inversion
(their parity). A simple example is the classification of the
orbitals of the hydrogen molecule: there is a point of inversion
symmetry of the molecule at the middle of the bond. Consider
the amplitude of the 1 sobonding orbital at an arbitrary point
of the molecule and project a straight line from this point
through the inversion centre, and travel an equal distance to
the other side (Fig. G2). The sign and magnitude of the
amplitude of the obond at the new point is unchanged, and so
the orbital is even (g) under this symmetry operation, and is
FIG. G2. Gerade and ungerade symmetry.
Grotrian diagram
written a . On the other hand, the same journey in the "anti
bonding orbital (1sff*) begins at a point of positive amplitude,
passes through a node, and ends at a point of negative
amplitude. This orbital is odd (u) and is labelled a*. The 2p0
orbitals behave similarly, but the TTorbitals behave differently
(see Fig. G2): the bonding TTorbital is u and the antibonding
orbital is g.
The g, u classification also applies to states: the appropriate
label is obtained by determining whether the state is even or
odd overall by considering the product of the inversion
behaviour of its components. Thus H 2 is g because (even) X
(even) is even; the excited "configuration a a* is u because
g X u = u.
When there is no centre of inversion symmetry the classi
fication is inapplicable: g and u labels may be added to the
states and orbitals of homonuciear diatomics and to
centrosymmetric octahedral complexes, but not to hetero
nuciear diatomics or to tetrahedral complexes.
The classification is useful in a discussion of selection rules
because the only "electric dipole transitions allowed are those
involving a change of parity; thus g — ► u and u — »■ g transitions
are allowed, but g — > g and u — * u are forbidden.
Questions. What is the significance of the labels g and u ? Why
are they not applicable to heteronuctear diatomics? Which of
the following molecules could have orbitals and states dis
tinguished by g and u: 2 , NO, N0 2 ,C0 2 ,CH 4 ,He, NH 3 ?
Classify the following orbitals into g or u type symmetry:
s, p, d, forbitals on free atoms; Iso, 2p7T*, 3der, 3d7T, 3d7T*,
3d5 in homonuciear diatomics. Apply g and u labels to the
ground states of He, H 2 , and 2 . Which electric dipole
transitions are allowed under the g, u classification?
Further information. The g, u labelling is a grouptheoretical
classification of the parity of a state; therefore see MQM
Chapter 5, Cotton (1963), Bishop (1973), and Tinkham
(1964). Also see books on molecular structure, such as
Cou [son's The shape and structure of molecules (OCS 9),
Coulson (1961), and Murrell, Kettle, and Tedder (1965).
King (1964) and Herzberg (1950) both discuss the role of
g, u classification in the electronic spectra of molecules.
Grotrian diagram. In a Grotrian diagram the energy levels of
an atom are displayed as a ladder of lines classified into con
venient groups. Spectral transitions are represented by lines
connecting the "terms between which they take place, and the
frequency (or wave number) of the transition can be added if
desired. Sometimes the relative intensities of the transitions are
indicated by the thickness of the connecting lines. An example
of a very simple Grotrian diagram is shown in Fig. G3: this is
for the "hydrogen atom. The levels are classified according to
the principal "quantum number, and the classes correspond to
different values of the "orbital angular momentum quantum
number id. The different transitions giving rise to various
"series of lines are indicated.
14
IZ
10
E/eV
6
4
2
n = l
Bradierr Pfund
PtadM
20000
BohMi
Lyman
40000
■60000
80000
100000
FIG. G3. Grotrian diagram for atomic hydrogen.
* tonaotion limit ■*
Enenjy
cm" 1
 5000
10000
IS 000
20000
25000
SJ0O0
35000
40000
FIG. G4. Grotrian diagram for atomic sodium.
The Grotrian diagram for sodium is shown in Fig. G4. When
there is more than one electron confusion sometimes arises in
the labelling of the energy levels because in some cases the
orbital occupied by the excited electron is used as the label,
and in others the term symbol, the label of the state of the
atom. Thus in some cases the lowest state of the sodium atom
is labelled 3s, in others it is labelled 3s 2 S, in others 3 2 S, and in
others, reasonably but dangerously, 1 2 S, denoting that the
ground state is the first of the doublet states. It is the pen
ultimate of these labels that we shall employ.
group theory 87
The energy is generally referred to the ground state as
zero. Fragments of the diagrams may be magnified if it is
desired to display the "fine structure.
Further information. See MQM Chapter 8 for further examples
and further discussion. Grotrian diagrams are discussed in King
(1964), Candler (1964), who gives many examples, Kuhn
(1962), and Condon and Shortley (1963). A collection of them
is given in the American Institute of Physics handbook,
p. 712, Gray (1972). The standard collection of energy level
data for atoms is that of Moore (1949 et seq.). Other aspects
of the use of the diagrams is described under "fine structure,
series, and the "hydrogen atom.
group theory. In quantum theory, group theory is the
mathematical theory of symmetry. It puts on a formal
mathematical basis our intuitive notions about the symmetry
of objects, and so enables unambiguous deductions to be
drawn about the consequences of their symmetry. Taking
full advantage of the symmetry of a system reduces the
amount of labour involved in calculations, and it often enables
conclusions to be drawn without the need for elaborate
manipulations.
Group theory is used to find the appropriate linear
combination of atomic orbitals for the "molecular orbitals of a
molecule, to classify atomic and molecular states, to determine
the "selection rules that govern the transitions between these
states,. and to find the normal modes of "vibration of mole
cules. Angularmomentum theory can be regarded as a
branch of group theory. The language of group theory is
concerned with symmetry transformations of molecules and
solids: with their "matrix representations, with their character
and their class, and with the manipulation of the irreducible
representations.
A brief summary of the properties of characters, which are
the most useful aspects in most chemical applications of group
theory, is given in Box 4 on p 33 for those who already know
some theory.
Further information. See MQM Chapter 5 for a discussion of
basic group theory, and the remainder of the book for extensive
applications. Its connexion with angularmomentum theory is
88 group theory
discussed at the end of Chapter 6 in MQM. Helpful elementary Hamermesh (1962), Weyl (1930), and Judd (1963). An
introductions to group theory are the books by Jaffe and exhaustive treatment of the symmetry properties of solids has
Orchin (1965) and Cotton (1963), and an introductory book been prepared by Bradley and Cracknell (1972), and a small
with many of the details filled in is Bishop (1973), Other handbook containing character tables and group properties to
helpfuf introductions include those by Tinkham (1964), act as a notebook for group theory calculations has been
Schonland (1965), McWeeny (1963), and, for solids, Wooster prepared by Atkins, Child, and Phillips (1970). An entertain
(1973), For advanced treatments see Wigner (1959), ment has been written by Weyl (1952).
H
hamiltonian. The name of Sir William Rowan Hamilton
(1805—65), who was Astronomer Royal of Ireland as an
undergraduate, was a devotee of pork chops, but not the
husband of her who entertained Horatio Lord Nelson, is
commemorated in quantum mechanics because he set up a
system of classical mechanics ideally suited to the structure of
quantum mechanics, and because he almost discovered
quantum mechanics itself, and quite possibly would have done
if during his life experiment had required it. For the systems
that concern us the hamiltonian is the sum of the kinetic T
and potential V energies (but like most things, it can be
defined more generally, more subtly, and more powerfully).
We give it the symbol H (some prefer 30 and so write
H = 7+ y. In quantum mechanics it is necessary to interpret
observables as "operators, and interpreting the observables 7"
and V as operators we see that the hamiltonian is the operator
for the total energy of the system. The total hamiltonian for a
system consists of a sum of terms corresponding to different
contributions to the total energy. A selection of such terms is
listed in Box 7 (overleaf).
The Schrodinger equation is often written in the form
H\(j= E4>, and so it can be interpreted as an eigenvalue
equation, the energy E of the system being the eigenvalue of
the hamiltonian operator, and the wavef unctions the corre
sponding eigenstates.
Questions. What is a hamiltonian? What is a hamiltonian
operator? How may the Schrodinger equation be expressed in
terms of the hamiltonian, and how may it be interpreted?
Write the hamiltonian, and analyse the significance of each
term, for the following species: a free electron; a hydrogen
atom; a hydrogen moleculeion; a hydrogen molecule; a helium
atom; a carbon atom; an harmonic oscillator; an harmonically
oscillating charge in an electric field; a mass on the end of a
spring; and the hydrogen atom in a magnetic field.
Further information. See MQM Chapters 3 and 4 for an
introduction to simple ideas about hamiltonians, and Pauling
and Wilson (1935) for another discussion. All books on
quantum theory discuss hamiltonians, and for the classical
background see Goldstein (1950). For further information,
see operators. Most hamiltonians are very complicated, and
their eigenvalues cannot be found exactly: the three major
approximation techniques are "perturbation theory,
•variation theory, and selfconsistent field techniques. A
bibliographical note about Hamilton, and his behaviour at
breakfast, will be found in Scientific AmericanWhittaker
(1954).
harmonic oscillator. Harmonic oscillations occur in classical
mechanics when the restoring force on a body is proportional
to its displacement from equilibrium. A force —kx, where k is
the *forceconstant, implies the existence of a potential j**
(because F = 3 W3x). An harmonic oscillator in quantum
mechanics is a system with such a parabolic potential, and its
properties can be determined by solving the "Schrodinger
equation with a potential \kx 2 . The conclusions this leads to
are as follows.
1 . The energy is quantized and limited to the values
(v + jlhojc, where «o = ik/m)^ and v is confined to the
89
90
harmonic oscillator
In general
Free particle
BOX 7; Hamiltonians
tf= 7"+ V
#/
0<x<£,
H = {tf/2m)V 2
"Particle in a box (onedimensional square well)
W = (hV2«)l
"Hydrogen atom
W = {hV2m)V 2 e 2 /47re r.
Helium atom
H =  {h 2 /2m\^  (h 2 /2m}Vl  2e 2 /47i€ r 1 
 2e 2 /4we r 2 + e 2 /47reor I2 .
Hydrogen moleculeion (H^); fixed nuclei
H =  ih 2 /2mW 2 , e 2 /4m=,>r 1A e 2 /4ire r tB +
+ e 2 /47re r AB .
"Harmonic oscillator
H = ~ihV2m)(£^j +l*x 2
Rigid "rotor
H =  [h 2 l2l xx )f;  [h 2 i2t yy )J 2 y  ih 2 l2tjj].
"Magnetic dipole moment jU in field B
H ~ — MB; for example, ju = gy s or 7 I.
Electric dipole moment ju in field E
H = ~ii£.
values 0, 1 , 2, . . .. This implies the existence of a zeropoint
energy of jhWo when the oscillator is in its lowest energy state
with V= 0: all the energy cannot be removed from an
oscillator. The zeropoint energy may be viewed in the light of
the uncertainty principle: eliminating all energy, and therefore
momentum, would imply an infinite uncertainty in position;
but the particle is confined by the potential. The other
implication is that the spacing between energy levels is ha^,
and the energy of osciliation can be increased only by absorb
ing integral multiples of the vibrational "quantum iiGJo. The
consequences of this are discussed further in "heat capacity
and quantum.
2. The wavefunctions are simple polynomials in the dis
placement (the Hermite polynomials) multiplied by a gaussian
function. The explicit form of some of the functions is given
in Table 1 1 together with some of their more important
technical properties. The shapes of a few of them are drawn in
Fig. H1. In the lowest state the polynomial is simply the
trivial factor 1, and so the wavefunction is the bellshaped
gaussian curve which has its maximum at the equilibrium
position. As the "wavefunction gives the distribution of the
particle it follows that, in the ground state, the particle
clusters close to its equilibrium position but possesses both
"kinetic and potential energy by virtue of the wavef unction's
FIG. HI, The wavefunctions and energy levels of a harmonic oscillator.
The classically accessible domains are shown bv the strong lines confined
by the potential. View this in relation tn the upper half of Fig, F4.
harmonic oscillator
91
shape and the presence of the potential. The next poly
nomial is essentially the factor*, and so the product of this
and the gaussian is a wave with a node in the centre (at x — 0).
This corresponds to a higher energy because the wave is more
sharply curved (higher "kinetic energy) and penetrates more
deeply into the potential (in classical terms it swings both
faster and further). The next function again peaks in the
middle, but has significant accumulations of probability in
the regions of higher potential. As the excitation increases the
principal peaks of the probability distribution appear more
dominantly at the limits of the distribution, at what in the
classical treatment are the turning points of the oscillation.
This is in accord with the classical distribution, for at the
turning points the kinetic energy, and therefore the velocity,
is least, and the probability of finding the particle there the
greatest.
3, A "wave packet may be formed when the energy state of
the oscillator is imprecise. A wave packet with a gaussian shape
moves from one side of the well to the other with a frequency
COo in a manner that resembles the classical motion (Fig. H2);
classical motion ^__^
motion of wave packer
FIG. H2. Classical limit of the harmonic oscillator.
therefore we see that the quantum spacing cjq becomes the
frequency of the classical oscillator. This is a very good
example of the way that quantummechanical principles
underlie classical mechanics (see "correspondence principle).
An oscillator behaves more closely in accord with classical
mechanics as a "superposition of states becomes more
justifiable; this increases as the quantum separation decreases.
Therefore it can be appreciated that oscillators with fre
quencies of the order of 1 Hz (or 2w rad s" ! ), such as a
classical clock, behave essentially classically, whereas periodic
processes in atoms and molecules, with frequencies in the
range 10 12 10 15 Hz, behave quantummechanical ly.
The importance of the harmonic oscillator is based on a
number of features. First, oscillations in Nature are often
harmonic to a very good approximation, and therefore the
theory of the harmonic oscillator can be used in the descrip
tion of the vibration of molecules and of atoms in solids.
This leads on to its application in the theory of "heat
capacities and to many other properties of solids. Next, just
as in classical mechanics, the algebra of harmonic oscillations
is closely related to the algebra of "rotational motion, and
therefore it is not surprising to see harmonicoscillator algebra
appearing in some discussions of angular momentum. Finally,
the harmonic oscillator is a remarkably simple creature to deal
with because the expression for its energy is symmetrical in
the space and momentum coordinates (both are quadratic
functions). This feature makes it very simple to handle and is
responsible, for example, for the equal spacing of the energy
levels of the oscillator. The harmonicoscillator algebra is the
basis of the technique of "second quantization.
Questions. 1. Under what circumstances are oscillations
harmonic? What potential does this imply? What values of the
energy are permitted to the quantized oscillator? What is its
minimum permitted value? Why is it plausible that a zero
point energy exists for an harmonic oscillator? What is the
separation between any pair of adjacent levels? What is the
mathematical form of the wavefunctions? Discuss the form of
the distribution of displacements when the oscillator is in its
ground state, and compare it with the ground state of a
classical oscillator and with the state of a classical oscillator
containing the same amount of energy. How does the dis
92
heat capacity
tribution of displacements change as the oscillator is excited?
What is the motion of a suitablyformed wave packet? Under
what circumstances may such a packet be formed, and how
does it relate to the motion of a classical oscillator with the
same forceconstant and mass? Calculate the energy separation
for an oscillator of frequency 1 Hz, 10 l4 Hz (remember to
convert these frequencies to rad s"' by multiplying by 2?r).
Calculate the energy difference between 1 mo I of each kind of
oscillator in its ground and excited states, and express the
result in J mol"" 1 and in cm 1 .
2. Use the properties of the Hermite polynomials in Table 1 1
to deduce the selection rules for electric dipole transitions in
an harmonic oscillator. See °anharmonictty for a further
development of this question. Show that the harmonic
oscillator Schrodinger equation can be written in the form
a*a&
(X — 1 ) ^ where a is the differential operator
(d/dy) + y, a" is (d/dy>  y, y = (mtufe/h)* 1 *, x the dis
placement from equilibrium, and X is related to the energy by
E = X(h a^/2). The lowest energy corresponds to A = 1, and
so a a^i = 0; therefore a solution is a^j = 0. Solve this first
order differential equation, and show that it is indeed the
gaussian function in Table 11.
i
Further information. See MQM Chapter 3 for a discussion of
the properties of the harmonic oscillator and the solution of
the harmonicoscillator Schrodinger equation by the method
of factorization (in terms of annihilation and creation
operators), and see glll.11 A of Pauling and Wilson (1935) for
the solution by the polynomialexpansion method. The
oscillator solution is also described in Landau and Lifshitz
(1 958a), Schiff (1968), and Messiah (1 961 ). The properties of
the Hermite polynomials are listed in §22 of Abramowitz and
Stegun (1965), where numerical values will also be found; the
manipulation of the polynomials is described in §3. 10 of
Margenau and Murphy (1956). The relation of harmonic
oscillator algebra to angular momenta is described in Lipkin
(1965), Mattis (1965), Englefield (1972), and by Schwinger
(1965).
heat capacity. The theory of heat capacities of crystalline
solids began with Dulong and Petit's 'law' that all metals had a
heat capacity of 6 cal deg 1 mol"' (25 J K" 1 mol" 1 ). Unfor
tunately most metals do not have this heat capacity, and none
do at low temperatures. Nevertheless it is helpful to understand
the reasoning that 'justifies' Dulong and Petit assertion,
because the quantum theory is then more easily understood. A
block of metal contains N atoms, and each can vibrate against
its neighbours in three perpendicular directions; therefore the
block behaves like a collection of 3/V oscillators. The equipar
titton theorem states that with each oscillation at thermal
equilibrium can be associated an amount of energy kT\ there
fore the total energy of the block at the temperature T is
3NkT, or 3RT if the block is 1 mol of metal. Thermodynamics
tells us that the heat capacity at constant volume is {%UI'dT) v ,
where U is the internal energy of the sample. In our case
U= 3RT and so the heat capacity is C v = 3fl, the numerical
value of which accords with Dulong and Petit's rule.
Quantum theory warns us that the equipartion rule applies
to a classical system and may fail for systems that, like the
vibrations of atoms in crystals, ought to be treated by quantum
mechanics. The root of the discrepancy lies in the inability of
an oscillator to accept less than its full 'quantum of energy:
this has the effect of quenching the effectiveness of the
oscillators that constitute the sample, and therefore of lower
ing its heat capacity.
The Einstein model pretends that every oscillator in the
block has the same fundamental frequency V , and we shall
begin with this simplified version of the true situation. Imagine
a source of heat of temperature Fin contact with a collection
of oscillators all having the same frequency. If the oscillators be
haved classically each one would be activated when the sample
was in thermal equilibrium with the source. Each would swing
with its natural frequency, but with an amplitude such that its
mean energy was kT. But as the oscillator is governed by
quantum mechanics, if heat is transferred to the metal at a
low temperature it can be used to activate only a very small
number of oscillators, for no oscillator can possess an energy
less than hV E if it is to be excited at all. A little energy must
reside in a few oscillators, and the remainder must be
quiescent. Therefore there is an effective reduction in the
number of oscillators in the sample, and a consequent re
duction of its heat capacity. At higher temperatures the energy
may be distributed over many more oscillators and so more
heat capacity
93
BOX 8: Heat capacities of solids
Dulong and Petit
C y = 3Lk = 3R .
Einstein
C, = 3R
(O^' l expfl E /71 1
\T/ lf1 exp(e E /D] 2 /
6=hvjk.
Debye
V 3 r6„{T
~U2j 4 fl/56™)7 a T<8 D
d =hvjk.
J> D and B D may be related to the speed of sound v.
*£ = 3Nv 3 /4irV,
where NIV is the number density of atoms in the sample.
are able to accept energy: therefore the heat capacity is greater
than at lower temperatures. At very high temperatures, when
the energy of the block greatly exceeds the excitation energy
of the oscillators, all oscillators are effective, and may be
stimulated to high quantum levels, and the sample attains its
classical heat capacity of 3ff. Box 8 gives the Einstein
expression for the heat capacity, and Fig, H3 shows the
pred i c ted te m pe ratu rede pe nde n ce .
The Debye model is a modification of the Einstein model
and takes into account the fact that the oscillators have a
range of fundamental frequencies from zero up to a limit V .
One may understand this situation in two different ways. The
first way is to regard the solid as a continuum (as a jelly). The
jelly can vibrate at all frequencies from zero up to a very high
value and the number of modes of oscillation that have a given
frequency can be calculated quite simply. But the total number
of oscillatory modes cannot exceed the total number of
vibrational modes of the atoms that constitute the jelly. This
number is 3/V; therefore there must be an upper limit to the
frequency of vibration of the jelly such that the total number
of oscillators is equal to 2N. (If the jelly were a true con
30
FIG. H3. Calculated heat capacity
curves.
T/e
M
70
— i —
90
94
heat capacity
tinuum, like the vacuum, there would be no upper limit to the
frequencies because a continuum corresponds to a system with
an infinite number of oscillating components: for this reason
the vacuum can support all frequencies of light.}
FIG, H4. Oscillations in a chain of atoms, (a) the lowest frequency,
(b) an intermediate frequency, and (c) the highest frequency.
The other method of seeing that an upper frequency
limit must exist is to consider Fig. H4, which shows the chain
of atoms in a crystal vibrating relative to each other. If we
consider only transverse vibrations (those perpendicular to the
line of atoms) it should be clear that the vibration with the
highest restoring force is the one in which neighbouring atoms
are displaced in opposite directions, and the vibration with
least restoring force is the one where all the atoms are dis
placed in the same direction (but by different amounts; if
they were all displaced by the same amount we should obtain
a translation of the block). There will also be intermediate
modes of displacement, and therefore we can expect a range of
restoring forces, and so a range of fundamental frequencies;
but there will be a maximum frequency because it is impossible
to obtain a higher restoring force than that in the situation
where the direction of displacement changes between neigh
bours. This situation is illustrated in Fig. H5.
The Debye model takes the distribution of oscillations into
account by assessing the number of fundamental oscillations
of each frequency v between and V Q and calculating the
NUO
FIG, H5. The Einstein, Debye, and experimental distribution (for Cu)
of the number of oscillators of a frequency v.
total contribution to the heat capacity (see Box 8 and Fig. H3).
Since there are oscillators at lower frequency than in the single
frequency Einstein model the heat capacity of the Oebye
model exceeds that of the latter, but the difference disappears
at high temperatures when the behaviour is virtually classical.
The heat capacity depends on the temperature and a charac
teristic constant known as the Debye temperature d = hv Ik;
since the cutoff frequency is higher in rigid materials so too is
the Debye temperature. Some representative values of 6 are
listed in Table 12. A high Debye temperature tends to lower
the heat capacity at a given temperature. At low temperatures
the Debye model predicts that the beat capacity should be
proportional to (7V0 D ) 3 , and this is often employed in
entropy calculations.
Further information. Chapter 1 of MQM gives details of the
calculation of the Einstein heatcapacity formula, and some
information about the Debye calculation. For an account of
the Einstein calculation, and a discussion of Debye's, see
Gasser and Richards' Entropy and energy levels (OCS 19). A
very good discussion of heat capacities, with full details of
calculations, will be found in Chapter 6 of Kittel (1971) and
Chapter 2 of Dekker (1960). For further information consult
Chapter 16 of Davidson (1962). A compilation of numerical
HellmannFeynman theorem
95
data, including the numerical values of the Einstein and
Debye functions and the Debye temperatures of many
materials, will be found in §4 of Gray (1972). For the
Debye function also see §27 of Abramowitz and Stegun
(1965). Some Debye temperatures are given in the other
books mentioned.
HellmannFeynman theorem. The theorem states that
the gradient of the energy with respect to some parameter
6E/6P is equal to the expectation value of the gradient of
the hamiltonian with respect to that parameter, dH/dP. In
order to calculate dE/dP, where P is the parameter (which
might be a molecular bond length or angle, a nuclear charge,
or the strength of some applied field), it is sufficient first to
calculate dH/dP, which might turn out to be a very simple
operator, and then to ca leu fate its expectation value. The
simplicity of this statement conceals a severe limitation
(there has to be some catch): in this case it is the not
unreasonable requirement that the wavefunctions used to
calculate the expectation value must be exact. All the work
lies in their evaluation, and faint hearts often apply the
HellmannFeynman theorem even when only scruff ily
inaccurate functions are available. This misuse can introduce
significant errors when the wavefunctions are only slightly
bad; so beware.
When is the theorem used? One application is to the
calculation of the response of molecules to electric and
magnetic fields: see "polarizability. Another very interesting
application is to the study of the geometry and force
constants of molecules. This application stems from the
remarkable consequence of the theorem that the force on a
nucleus in a molecule may be calculated as a simple problem
in classical electrostatics if the exact (quantummechanical)
charge distribution is known. This deduction from the
theorem is often called the electrostatic HellmannFeynman
theorem. If we know the electron density everywhere, and
that is known if the "wavefunction is known, then the force
on a nucleus can be calculated by considering the
Coulombic force that the same classical charge distribution
exerts on a point nucleus. The •forceconstant for any
distortion of the molecule can then be calculated by
working out the restoring force on a nucleus when the
geometry is distorted. Note that this derivation treats the
equilibrium geometry as a problem for classical electrostatics;
but do not be misled into thinking that al! the nonsense about
•exchange energy, and so on, is an unnecessary nonsense after
all. The complication of exchange interactions is hidden, of
course, in the difficult task of determining the correct electron
density for a given nuclear configuration. Nevertheless, the
theorem does enable one to remove some of the mystery about
the shape of molecules, for in some sense the geometry can be
understood in terms of a balance of electrostatic forces, even
though the distribution responsible for the balance is governed
by quantum mechanics.
Questions. 1. State the HellmannFeynman theorem. Under
what conditions is it untrue? Propose several possibilities for
the parameter P in the calculation of various molecular
properties. What use may be made of the theorem in the dis
cussion of molecular structure? Why do problems of exchange
energy remain even though the molecular shape may be under
stood in terms of the forces exerted on the nuclei? Given
that the theorem enables the forces exerted on nuclei to be
calculated, what conditions on the forces lead to the determin
ation of the equilibrium geometry of the molecule? What
might the parameter P represent in a diatomic molecule?
What is the influence of the "exchange energy on the determin
ation of molecular shape according to the theorem?
2. Prove the HellmannFeynman theorem by considering the
expression E{P) ={\p(P)\ H {P)\ip{P)), where the energy depends
on the parameter Pand so do both the hamiltonian and the exact
normalized wavefunctions (so that (\p{P)\\{/lP)) = 1 ). Differ
entiate both sides with respect to P and use the fact that
4>(P) is the eigenfunction of H(P) with eigenvalue E[P\.
Further information. A simple account of the theorem in its
application to electric and magnetic problems is given in MQM
Chapter 11. For molecular applications see a thorough review
of the subject by Deb (1973), where many more references
will be found. More information is given in §2.6 of Slater
(1963). For the original exposition of the theorem see
Hellmann (1937) and Feynman (1939).
hermitian operators. An "operator ffj is hermitian if the
integral /d7f*S2ff is equal to the integral /dr(iV) *g. In the
Huckel method
Dirac "bracket notation the requirement would be for
(fKllg) to equal {g\£l \f>*. Hermitian operators are important
in quantum theory because their "eigenvalues are real; there
fore operators corresponding to physical observables must be
hermitian. Another consequence of hermiticity is the
orthogonality of "eigenf unctions corresponding to different
"eigenvalues of hermitian operators.
Questions. 1. What is the meaning of 'hermiticity'? What
properties stem from the hermitian nature of an operator, and
why are hermitian operators important in quantum mechanics?
Is the operator 'multiply by x' hermitian? Is the operator d/dx
hermitian? Is the operator (h/i)(d/dx) hermitian? (In investi
gating the hermiticity of d/dx use integration by parts and the
property that the functions f and g disappear at sufficiently
distant boundaries,} Which of these operators might correspond
to what physical observables?
2. Prove that the eigenvalues of an hermitian operator are real,
and that eigenf unctions corresponding to different eigenvalues
are "orthogonal. Demonstrate these results specifically in the
case of the operator fi ? = (h/iH3/30) which occurs in the
theory of "angular momentum.
Further information. See MOM Chapter 4 for a further dis
cussion of hermiticity, its consequencies, and the proof of the
properties mentioned. See also Dirac (1958), von Neumann
(1955), Jordan (1969), and Jauch (1968) for detailed accounts
of the quantummechanical aspects of hermiticity.
Huckel method. By taking into account the symmetry of a
molecule in a wise way, and by making foul assumptions that
no referee would pass nor examiner condone, Huckel devel
oped a simple scheme for calculating the energy of the it
electrons in conjugated systems. The approximations, which
we label 1—6, are extreme, for they neglect every complicated
aspect of the exact problem.
1. The erelectrons are ignored; they are present implicitly
because they are largely responsible for determining the shape
of the molecule, but their interaction with the ff electrons is
neglected.
2. All overlap is neglected. Since the overlap integral
between neighbouring carbon TTorbitals is about 0'25 the
propriety of this approximation can be appreciated. (In the
event it turns out that overlap can be included quite simply,
and It does not affect the answers savagely.)
What about all the complicated integrals that occur in the
exact problem? They are classified as too difficult; but to
eliminate all integrals would eliminate the problem. Therefore;
3. All the integrals involving only one atom (roughly
corresponding to the energy of an electron occupying a
carbon 2porbital) are set equal to a: this is called the
'Coulomb integral.
4. All the integrals involving atoms separated by more
than one bond are ignored as far too difficult.
5. All the integrals involving neighbouring atoms are
set equal to the same value (3: this is called the resonance
integral.
6. After all these approximations it does not seem worth
"antisymmetrizing the wavef unction; so it isn't. (But it
should be realized that in a rather illdefined sense the effect
of antisymmetrtzation— the role of "exchange energy— has
been taken into account by the parameters a and j3.)
With these approximations in hand the "variation
method is applied to determine the best "linear combination
of atomic norbitals to describe the structure of the mole
cule; this leads to a "secular determinant whose roots give
the energy of the orbitals. (Each diagonal element of the
determinant is a— £ and every offdiagonal element is zero
except those corresponding to neighbouring atoms, which
are set equal to p\) When the secular determinant is solved
and the molecularorbital energies are known the
coefficients for the atomic orbitals may be discovered.
This is essentially the complete solution to the problem in
the Huckel approximation, and it is possible to deduce a
number of molecular properties. These include the
derealization or "resonance energy, the energy of elec
tronic transitions within the Trsystem, the "charge density
on the carbon atoms, and the TTeiectron contribution to
the dipole moment, the "bond order, and the "free
valence. In calculations of derealization energy a reason
able value of is — 069 eV (—67 kJ mol" 1 ), but for spectral
transitions a better value is —271 eV (—21 900 cm 1 ). A
sample calculation is illustrated in Box 9.
Huckel method
97
BOX 9: Huckel calculation; an example
In general:
1. Write MO as ^ = Lc.0..
2. Set up secular determinant' \H.. — ES.. I as follows:
M if /=/
S/ i~ to if /*/
!a if / =/
j3 if (¥*},
otherwi?
but/, /are neighbours
otherwise,
3. Solve \H..  ESJ = for energies E.
For butadiene $ = C i0j f c 2 4>2 + c 3 3 + c 4 4 , where
0i, ... 04 are the 2pTTorbitals on the carbon atoms in
CH 2 :CHCH:CH2, The secular determinant is
cHF
af 3
o£
0/3 oF
If x/J = a— f this reduces to x 4 — 3x 2 + 1=0. Therefore
the roots are
£ = a+160 and £ = af±O60
and these are the energies of the four molecular orbitals of
butadiene. The secular determinant can usually be simplified
(factorized) by using symmetry arguments. See Further
information.
=
The Huckel method is moderately satisfactory because
of its reliance on the symmetry of the molecule: the
orbitals are essentially classified according to the it
symmetry, and then the number of bonding and anti
bonding juxtapositions of overlapping 7rorbitats is counted
by the coefficient of j3; in this way one obtains a rough
guide to the ordering of the energies.
There is obviously enormous room for improvement in the
method, and an enormous amount of work has been done with
that in mind, A simple improvement is the inclusion of overlap
between neighbours: this squashes the lower, bonding orbitals
together and separates the upper, "antibonding levels without
significantly affecting their order. The next improvement often
employed in simple calculations is to realize that the energy of
an electron on a carbon atom, measured by the magnitude of
a, depends on the charge density on that atom. The Huckel
method ignores this dependence and gives the same value of
a to all the atoms irrespective of the focal accumulation of
electron density. (See "alternant hydrocarbon in this
connexion.) The a>technique seeks to repair this deficiency
by making a depend, through a constant of proportionality Co,
on the charge density. This is a simple example of a "self
consistent calculation: a Huckel calculation is first done to
find the charge density, then each a is modified appropriately,
and the calculation repeated with the new set of as: the
scheme is repeated until the charge density remains constant
through a cycle of the calculation.
Beyond these trivial modifications of the original theory
one encounters the semiempirical methods which are at the
centre of much of presentday research. These attempt to
relax the bold assumptions of the Huckel method and to
approach the accuracy of an exact calculation. They proceed
by neglecting electronrepulsion integrals in a moreorless
rational fashion, and expressing the magnitude of the remain
ing molecular integrals in terms of some empirical quantity, or
in terms of quantities calculated for atoms, or leaving them as
adjustable parameters. Virtually all of them deal with the
valence electrons only, and acknowledge the existence of the
core electrons only in terms of the final choice of the para
meters. Most do not make the Huckel distinction between the
a and 7Telectrons except in so far as their symmetry is con
cerned; therefore the methods can be applied to molecules
lacking planes of symmetry. Popular methods at one time
included the PariserParrPople (or PPP) method, which it only
a slight improvement on the Huckel method and ignores
virtually all the electron repulsion integrals between atoms.
The CNDO {complete neglect of differential overlap) method
neglects fewer integrals and is the basis of many modern
calculations. Other initials, such as IN DO (intermediate
neglect of differential overlap) and MINDO (modified INDO),
represent schemes of neglecting different integrals and
Hund coupling cases
choosing the value of those remaining, Details of the actual
choices and approximations in these schemes will be found
under Further information.
Questions. 1. List the approximations that constitute the
Huckel scheme for calculating 7Telectron energies. What is the
significance of the parameters a and S? Why should a be
expected to depend on the charge of the atom? What mole
cular properties may be calculated in the Huckel scheme?
What improvements may be made quite simply? What
improvement is it much more difficult to introduce? What is
the major limitation on the accuracy of the semiempirical
methods? Why are they inferior to the best HartreeFock
"selfconsistent field calculations?
2. Using the calculational scheme set out in Box 9, set up and
solve the Huckel equations for the molecules CH 2 =CH— CH=
CH^, CH 2 =C=CH 2 , cyclopropene, and benzene, and calculate
the delocalization energy and charge distribution in each. In
the case of cy do butadiene investigate the effect of including
overlap between nearest neighbours: let the overlap integral
be S in each case (set S = 025 at the end of the calculation).
3. The tjtechnique supposes that the Coulomb integral on
atom r is related to the charge density g on r by the formula
a r = a + {1 —g) OJp 1 , with the coefficient oj being about 1 4.
Apply the technique to the ally I cation starting with the
Huckel approximation and proceeding through three cycles of
theo>method. What is the role of the G>technique in the case
of "alternant hydrocarbons?
Further information. See MQM Chapter 9. Many books deal
with the Huckel method in detail, and work through many
examples. See Coulson's The shape and structure of molecules
<OCS9)and §9.6 of Coulson (1961). See also Murretl, Kettle,
and Tedder (1965), Streitweiser (1961), Pilar (1968), Salem
(1966), McGlynn, Vanquickenborne, Kinoshita, and Carroll
(1972), and Dewar (1969). Accounts of the developments of
the theory will be found in these books and also in Pople and
Beveridge (1970), Doggett (1972), and Murrel! and Harget
(1972). A compilation of the results of such calculations has
been prepared by Coulson and Streitweiser (1965).
Streitweiser (1961) works through many calculations on
organic molecules, and shows how the results may be applied
to the prediction of chemical properties. Roberts (19616)
works through the bare bones of the method. Parr (1963)
has reviewed the modern theories of molecular structure, and
his book contains a collection of some of the important
original papers.
Hund Coupling cases. In a diatomic molecule there are
several sources of angular momentum, notably the "spin of the
electrons, their "orbital motion about the axis, and the
•rotation of the nuclear framework. The total angular
momentum is the vector sum of all these momenta, but there
is a variety of ways of coupling them together. The Hund
cases are a sensible selection of a few of the possible ways of
performing the coupling; we shall deal with the four simplest
and most common schemes.
(a) In Hund'scase (a) the strong electrostatic effect of the
nuclei is respected: the axial field allows only circulation about
the axis to survive and be a source of orbital angular
momentum. Through the "spin orbit coupling any resultant
spin of the electrons is also coupled to the surviving com
ponent of the orbital momentum; that is, the spins are coupled
to the axis by a twostep process. All the electronic angular
momentum is along the axis: the orbital component is Afiand
the spin component Sh giving a total momentum along the
axis of S2h with £1 = A + 2. But the nuclear framework is also
rotating with a momentum which can be represented by a
vector O perpendicular to the axis (Fig. H6 a). It follows that
the total angular momentum of the molecule is represented by
the resultant vector J.
(b) Hund'scase (bj deals with the situation in which the
coupling of the spin momentum to the axis disappears. This
may occur in the case where there is no electronic orbital
angular momentum (so there is no guidance via the spinorbit
interaction for the alignment of the spin) or when the spin
orbit coupling is so weak that the orbital angular momentum
does not succeed in fastening it to the axis. This situation may
arise in the diatomic molecules formed from the firstrow
atoms, especially in their hydrides. The orbital momentum is
still coupled electrostatically to the axis, but the spin swims
loosely around. Therefore the electronic orbital momentum
Hund coupling cases
99
FIG. H6. The Hund coupling cases (a)(d),
and the nuciear framework momentum couple into a
resultant N, and only then, in order to get the total
momentum of the system, is the spin coupled in to give a
resultant J (see Fig. H6 b).
(c) In Hand's case (c) we go to the other extreme and
encounter a situation in which the spinorbit coupling is so
strong that the spin and orbital momenta couple strongly at
every opportunity. This passion couples them into a resultant
total electronic angular momentum J . This J "processes around
the internuclear axis, on which it has a component £2h. To
this resultant couples the rigid rotation of the nuclear frame
work, and the whole system yields a total momentum J
(Fig. H6 c). The spinorbit interaction is strong in heavy
atoms, and it is for these that case (c) is important.
(d) In Hund's case (d) we encounter a peculiar situation in
which the electrons are virtually independent of the orient
ation of the nuclear framework. The pair of nuclei churn
around inside the molecule but the outermost electrons do
not respond. Such a situation arises when the electron has
been excited to an orbital outside the valence shell (into a
"Rydberg state), for it is then so far away that the pair of
nuclei resemble a single point nucleus. In this case the
electronic orbital angular momentum vector combines with
the nuclearframework vector to yield a resultant N; with this
combines the spin (there is very little spinorbit coupling in
such diffuse states) to give the total resultant J (see Fig.
H6d).
It should be clear that these four cases are only a few of the
many possible; and even more are possible if the nuclei them
selves possess "spin. Furthermore, the cases are extreme, ideal,
or pure cases; in any real molecule there are various com
petitions between different angular momenta, and for none
can there be complete victory. The contamination of one of
the pure Hund's cases by another is referred to as a decoupling.
Questions. What different sources of angular momentum exist
in a diatomic molecule? Under what circumstances does
Hund's case (a) dominate? What happens when there is a
vanishing orbital angular momentum in the molecule? What is
an appropriate coupling scheme when the spinorbit coupling
is small? What is appropriate when it is large? What scheme is
appropriate when an electron occupies a very diffuse orbital?
What is actually meant by the term 'coupling'? (Consider what
a particular coupling scheme represents in terms of energy;
review precession.) The quantum numbers Aand Z represent
welldefined situations in case (a) (they are good quantum
numbers); is that also true in case (c)? What of the quantum
number 12? What do these quantum numbers represent?
Further information. See MOM Chapter 1 for another look at
the coupling schemes. See Barrow (1962), King (1964), and
Herzberg (1950) for an account of the way the coupling cases
are employed. There are a few more cases of interest: find
them all splendidly discussed in Chapter 5 of Herzberg
100
Hund rules
(1950). One important application is in the discussion of the
•selection rules that govern electronic transitions in diatomic
molecules.
Hund rules. The Hund rules provide a simple guide to the
ordering of the energies of atomic states. They state the
following:
(1) for a given "configuration the nerm with the highest
"multiplicity lies lowest in energy;
(2) for a given configuration and multiplicity the term with
the highest 'orbital angular momentum lies lowest in energy;
(3) for a given configuration, multiplicity, and orbital
angular momentum the "level with the lowest value of the
total angular momentum J lies lowest in energy if the con
figuration represents a shell less than halffilled; and the state
with the largest value of J lies lowest if the shell is more than
half full.
As an example consider the configuration 1 s 2 2s 3 2p 2 and
the terms 3 D, ' P, and ' S to wh ich it gives rise. What is the
order of energies? Application of the first rule yields the order
3 D< ('P, 'S). Application of the second yields 'P< *S, and
so we can conclude that the terms should lie in the order
3 D < 'P < l S. The third rule enables us to order the three
levels of the triplet term ( 3 D). Since L = 2 in 3 D, and 5=1,
the levels correspond to J — 3, 2, 1 . Since the pshell is less
than half full (it can hold six electrons) the appropriate order
is Dj < D 2 < 3 D 3( and so we have a complete ordering.
The basis of the rules lies in electrostatic Interactions for
the first two, and magnetic interactions for the third.
1. The first rule has an explanation that is rather more
subtle than is found in most books. First, we can say that if
their spins are parallel, the electrons must be in different
orbital s ("Pauli principle), and therefore must be further apart
than if they were crammed into the same orbital; their
repulsion will be diminished on this account. But if we have a
configuration 9 x 9 y . where the electrons are in different
orbitals, it is possible for them to have either parallel or
antiparallel spins: which lies lower, and why? The con
ventional, mythical explanation of why the triplet (parallel)
configuration lies lower is that, because electrons with
parallel spins tend to keep apart {see "spin correlation),
their repulsion is less than in a singlet (paired) configuration.
This is not so. Detailed calculation on some cases has shown
that the repulsion between the electrons is greater in triplet
states than in singlets, and that the lowering of the energy is
due to the modification of the electronnucleus interaction.
Thus, in a triplet atom the electron distribution contracts, and
is stabilized by the improved nuclear attraction: the electron
repulsion rises because the electrons are closer together, but
this increase does not defeat the improvement in the nuclear
attraction. Presumably the "spin correlation helps to stop the
electron electron repulsion rising faster than the nuclear
attraction.
2. The second rule reflects the tendency for electrons
to stay apart if their orbital angular momentum carries them
in the same direction. Electrons circulating in the same
direction, and therefore leading to a large total orbital
angular momentum, can stay apart; but electrons orbiting in
opposition will meet frequently, and so have a large
Coulombic repulsive interaction.
3. The third rule is of magnetic origin because the order of
levels is determined by the 9 spinorbit coupling interaction.
When the spin "magnetic moment is opposed to the orbital
moment the magnetic energy is least; but such an arrangement
of moments implies that the two momenta are also in op
position, which corresponds to a low total angular momentum
(see "fine structure, and especially Fig. F2 on p. 74). The
inversion of the levels when the shell is more than half full re
flects the change in the sign of the spinorbit coupling constant.
Questions. State the three Hund rules. Which depend on
electrostatic interactions and which on magnetic? In which
does the spinorbit interaction play an important role? What
is the reasoning that explains the first rule? And the second?
And the third? Why does an inversion of the levels occur when
the shell is more then half full? Put the following terms in
order of increasing energy: 2 S, 2 P, 4 S, 4 D. Order their levels
appropriately on the assumption that they arise from a more
than halffilled shell.
Further information. See MQM Chapter 8 for a discussion. The
Hund rules are discussed by Kauzmann (1957), King (1964),
and Herzberg (1944). Tables of term and level energies have
hybridization
101
been prepared by Moore (1949 et seq,). The rules, of course,
sometimes fail: this may be due to "configuration interaction,
where the presence of another configuration depresses a term
below the position where the rules predict it should lie. The
view that conventional explanations of the first rule are all
wrong (including that in MQM) is based on work of Lemberger
and Pauncz (1970), Katriei (1972), and Colpa and I slip (1973).
See Walker and Waber (1973) for a modification to the rules in
the case of //coupling.
hybridization. If an electron occupies an "orbital that has
mixed s, p, d, . . . character on an atom, it is said to occupy a
hybrid orbital, and the process of forming that orbital is
known as hybridization. An sphybrid orbital, for example, is
one composed of equal proportions of s and pcharacter, and
the electron that occupies it may be considered to have 50 per
cent scharacter and 50 per cent pcharacter.
What does a hybrid orbital look like? We know that an s
orbital may be regarded as a spherically symmetrical standing
wave, and that a porbital is a wave with two regions, one with
a positive amplitude and one with a negative amplitude. The
hybrid orbital is a "superposition of these two standing waves,
and where their amplitudes are both positive there is con
structive interference, and where one is positive and the other
negative there is more or less complete destructive interference.
The superposition is lopsided, and as shown in Fig. H7 a. An
sp hybridized orbital contains 33 per cent scharacter, and an
sp 3 orbital 25 per cent. These orbitals are also illustrated in the
figure, and it should be clear that the amount of topsidedness
increases through the series. Note how the node shifts as the
hybridization changes (it passes through the nucleus in the case
of an unhybridized porbital, but not for the hybrids because
of their scomponent).
What is the point of hybridization? It will occur in mole
cules if there results a reduction in the energy of the molecule.
Is the energy of a molecule reduced when its atoms hybridize?
On first glance it might be thought unlikely. Consider, for
example, the carbon atom, in its ground state its "configuration
is s 2 p 2 (we ignore the deep 1selectrons), and the 2selectrons
lie some way in energy below the 2p. It is natural to think that
carbon will form bonds with its 2porbitals and be divalent. If,
FIG. H7. sp hybrid orbitals. Computed con lours for hydrogenic 2s,
2porbita!s. Note that the nodal surface is shifted from the nucleus.
102
hybridization
however, we can find enough energy to promote one of the
seiectrons into the pshell the atom attains the configuration
sp 3 : if each of the four s and porbitals hybridize to form four
sp 3 hybrids we can envisage a situation in which each electron
occupies one of the hybrids. The form of these is dear from
Fig. H7: they are strongly directional, and a little calculation
shows that they are directed towards the four apices of a
tetrahedron (Fig. H8) (see "equivalent orbitals). Nevertheless
we still have not recovered our initial energy investment which
was used to promote one of the selectrons. At this point we
can appreciate that where we shall recover the energy is in the
four bonds that we are able to form. Because of the strong
directional properties of the hybrids they will have a very
efficient overlap with the four neighbouring atoms, and we
can expect four strong bonds. This is where we draw our
reward: more than the initial promotional energy is regained
by the formation of four strong (and 'equivalent) bonds.
FIG, H8. Tetrahedral sp hybridization (for example, in CH4I.
The case of ammonia (NH 3 ) helps us to appreciate another
source of improvement in energy (Fig. H9). The nitrogen
atom has the configuration s 2 p 3 , and so we can expect to
form a pyramidal ammonia molecule by attaching one
hydrogen atom to each halffilled nitrogen 2porbital. Suppose
an selectron were promoted into the pshell; would the energy
FIG. H9, Ammonia and hybridization; the lone pair is drawn in colour.
In (a) there is no promotional energy, but high repulsive interactions
and moderate overlap; in (c) low repulsive interactions, strong overlap,
but high promotional energy; in (b) is illustrated the compromise. Only
two bonds are illustrated.
of the molecule be lowered? First we invest promotion energy
by agreeing to treat all the s and porbitals on the same
footing. We form sp 3 hybrid orbitals with a tetrahedral dis
position and strong directional properties. Then we feed in five
electrons and add on three hydrogen atoms. The resulting struc
ture consists of three 0bonds, each formed from a strongly over
lapping sp 9 hybrid and a hydrogen Isorbital (and each con
taining two electrons), and a further two electrons to form a
"lone pair. With three of the bonds we get a return on the
hydrogen atom
103
energy invested because of the excellent overlap; but we also
recover some energy from the lone pair, because it is con
centrated in a region where its electrostatic repulsive inter
action with the three rjbonds Is minimal; furthermore, the
larger bond angles yield a reduced HH repulsion. The actual
structure of ammonia is a compromise between the amount of
energy required to promote an selectron into a porbital, and
the energy that can be obtained by improving overlap and
diminishing nonbonding interactions. In the event it turns out
that the molecule is some way between the two extreme
structures. {This can be determined by measuring the bond
angles: in the pure porbital case the HNH angle is 90°, and
in the pure sp 3 orbital case it is 109° 28'; the experimental
value is 103°.)
Carbon is well suited for hybridization because its pro
motional energy is quite small (the promoted selectron can
enter an empty porbital); therefore it shows a range of
hybrids, depending on the demands made under the reaction
conditions. In CH 4 and its homologues the hybridization is
almost pure sp 3 ; in alkenes and in aromatic molecules there is
a planar, triangular array of sp 2 hybrids and the remaining
2porbital of this promoted state forms the jrbond. In
acetylenic compounds (alkynes) the fjbond is sphybridized,
the remaining two 2porbitals forming the 7Tbonds.
A list of the symmetries of hybrid orbitals is given in
Table 13. A discussion of the formation of hybrid orbitals will
be found under "valence state.
Questions. 1. What is meant by the term 'hybridization?
Sketch some hybrid orbitals {especially sp, sp 2 , sp 3 , sd, sd 2 ).
Why can hybridization lower the energy of a molecule?
Describe the formation of CH 4 , NH 3 ,and H 2 0, and explore
the contributions to the energy which are modified by
hybridization of the central atom. Why is hybridization of
special importance in carbon?
2. The four sp 3 hybrid orbitals may be expressed as
s + p x +p k + p,;s
P z ;sp x
p "+" p ;
s p x + fV ~~ p ?" Snow tnat tnese are directed towards the
apices of a regular tetrahedron, and that they are mutually
orthogonal. Express a hybrid orbital in the form
as + (3{&p x + mp + np z ), where £, m,n are the direction
cosines of the bond that we hope to form, and a 2 + ft 1 = 1 .
Take another equivalent orbital pointing along the direction
$.', m', ri and otherwise of the same composition. From. the
condition that the functions are orthogonal and normalized
deduce that a. and j3 are related to the angle between the
bonds by a 2 i$ 2 =  cos 0, and 1/0 2 = 2 sin 2 ^3. Discuss the
form of the hybrids when 8 is 180°, 90°, and the angle of a
tetrahedron. Plot scharacter as a function of 8. Show that the
effect of hybridization is important only when the orbitals
involved are of approximately the same size. Proceed by
writing a and in the preceeding expression as sin£ and cos£
respectively (so that a 2 + fi 2 = 1 is satisfied automatically),
and write Hp x + mp + np. as the orbital p. Find an expres
sion for the overlap with a neighbouring orbital tpin terms
of the overlap integrals S ■ <s$ and S — <p^>), and find the
expression for the maximum value of S as a function of £. The
answerisS = [S 2 + S I ) w . This shows that S differs
max s p
appreciably from S ot S only if S ~ S ; in the event of
. » P ' s,, P
these overlaps being equal 5 = 2 A S .
Further information. See MQM Chapter 9, and especially
Appendix 9,2 where a fuller discussion of hybridization,
equivalent orbitals, and localized orbitals is given, together
with the solution to Question 2. The importance of
hybridization in molecular structure is described by
Coulson in The shape and structure of molecules (OCS 9) and
in Chapter 8 of Coulson (1961). See also Murrell, Kettle, and
Tedder (1965), Pilar (1968), Kauzmann (1957), and McGlynn,
Vanquickenborne, Kinoshita, and Carroll (1972), The last
book describes the formation of hybrids in some detail and
evaluates a number of useful overlap integrals. The group
theoretical formation of hybrid orbitals is described in MQM
Chapter 9.
hydrogen atom. The hydrogen atom, consisting of an
electron surrounding a proton, was one of the fences that
classical mechanics failed to take, and one of the remarkable
successes of quantum mechanics and its later developments.
The spectral observations on the hydrogen atom showed
that it emitted and absorbed light at a series of welldefined
frequencies (Fig. H10). In 1885 Baimer spotted a relation
satisfied by the frequencies that lie in the visible region of the
spectrum. This Baimer series (Fig. H10) fitted the formula
104 hydrogen atom
1 05 000—
too 000
95 000 J
90 00C
85 000
25 0CO
20 000
I50O0
10 000—
5 000
■ 97253 
(02572
I2I567 lL,
Lyman
43*05 
4860 
feSfc.28 K
H.
Balmer
(visiMe region)
820 4 ,
0938 —
I28I8 
I4584 —
Ftosehen
I875J — '
—22788 
^262 r '
Bracket?
405I2
—74000 .
ipfjnd
FIG. H10. The spectrum of atomic hydrogen.
V= ff(1/2 2  Mn\), with n i an integer greater than 2. It is
tempting to speculate on the existence of other series of lines
in which 2 2 is replaced by n\,n 2 another integer, and in due
course Lyman discovered his series {in 1914 and the ultra
violet) correspondingtor^ = ],Paschen his [n 2 = 3; in 1908
and the infrared), Brackett his (n 2 = 4, in 1922 and the
infrared), and Pfund his (n 2  5, in 1924 and the far infrared).
Those, for the moment, are the facts.
°Bohr constructed a theory of the hydrogenatom structure
which drew on Rutherford's nuclear model and the quantum
hypothesis; but the model had defects, and it was replaced by
a deduction of the structure from the "Schrodinger equation
by Schrddinger himself in 1926. A principal feature of this
theory is that the energy of the atom is "quantized and limited
to values given by —R in 2 , where n is an integer greater than
zero (the principal "quantum number) and R the "Rydberg
constant, which is a collection of fundamental constants
having the value 109 677 cm" 1 . A transition from the level rti
to the level n 2 involves an energy change in splendid accord
with observation (see the Grotrian diagram. Fig. G3 on p. 86).
The features of the structure are as follows:
1. The electron is distributed around the nucleus in
"orbital s. Each atomic orbital can be distinguished by a set of
three "quantum numbers. These are the principal quantum
number n, the 'angularmomentum quantum number
(occasionally called the azimuthai quantum number) 9., and the
magnetic quantum number m^. The principal quantum number
may take any integral value greater than zero, the azimuthai
quantum number may take any integral value from zero to
n — 1, and the magnetic quantum number can take any
integral value between £ and — £. The energy of the state is
determined solely by the value of the principal quantum
number and is given by — R H ln 2 . (More concerning the
"Rydberg constant will be found under that heading.) The
angularmomentum quantum number determines the
magnitude of the "orbital angular momentum of the electron
about the nucleus through the formula [£(£ + 1)] ' A h. The
magnetic quantum number determines the orientation of this
angular momentum in space, and, in accord with the general
properties of "angular momentum, the component of angular
momentum of the electron about an arbitrarily selected axis is
equal to mnh.
hydrogen atom
105
color ifude
oiimufh
FIG, H1 1. The spherical coordinates used to discuss the position of the
electron in hydrogen.
2. The orbital corresponding to the state with quantum
numbers n, %, and m^ is in general a function that depends on
the distance r of the electron from the nucleus, the colatitude
(the angle away from the atom's north pole), and the azimuth
(the electron's longitude) {Fig. H11), and may therefore be
written $ n faJfi 0, (j>). This orbital function may be expressed
as a product of a function dependent solely on the radius and
of one dependent solely on the angles: \}/{r, 6, <j>) =
R ni^ Y 9jnJ^' $' The an 9ular functions Y^ (0$) are the
•spherical harmonics.
2(a) When & = the orbital is isotropic because V [6, 0)
is a constant (1/2JT* 4 ) and independent of the angles. The
orbital in this case is known as an sorbital, and an electron
that occupies it is distributed symmetrically around the
nucleus (Fig. HI 2). Since fi = the angular momentum of an
electron in this orbital is zero; and in this connexion it is also
important to note that there no angular "nodes in the orbital
(the number of such nodes determines the angular
momentum).
2(b) When £ = 1 the orbital is aporbital, and its angular
dependence is given by the function Y, „, (0, d>). This is not a
imp
constant, but has its maximum amplitude along particular
axes in space. When mg = the function is Y m (#, <j>), which
might look fearsome, but in fact is simply the function cosfl;
this has its extreme along the zaxis, where = or 180°, and
so an electron that occupies this orbital is most likely to be
found in regions concentrated along the zaxis {Fig. H 12):
FIG, H12. Representation (by approximate boundary surfaces) of
s, p, d, and forbitals.
106
hydrogen atom
for this reason it is referred to as ap^orbital. The other
possible values that m^ may have (when 8 = 1 ) are ± 1 , and in
both these the dependence on $ is as sinO: this implies that
both are concentrated predominantly in thexyplane and have
zero amplitude anywhere on the zaxis. Both Y x t and V,_i
are complex functions, {see Table 23 on p. 282), but the sums
and differences are real, and so these combinations are easier
to depict. y 1( i + Y } ., is the simple function sinfl cos0, and
so it is a standing wave concentrated along thexaxis, because
there both sin 6 and cos0 are maximal: this is ap orbital. The
other combination. K,, Y \,i is the function sin0sin0,
which is a standing wave concentrated along the /axis, and
therefore is referred to as a p orbital. These have the same
shape asp^, but differ in orientation (Fig. HI 2). With each
porbital there is associated an angular momentum of
magnitude [1(1 + 1)f /j ri,or (/ 2) h, and it should be clear
how the different values of the quantum number m« dis
tinguish the different orientations of the distributions of the
electron, and therefore the different orientations of the
"orbital angular momentum of the electron occupying that
orbital (for example, when m^ — ± 1 the electron is largely in
the xyplane and so most of its angular momentum is about
the zaxis).
2(c) Next up the scale of S values lie the dorbitals with
£ = 2. Of these there are five (mn = 2, 1, 0, —1, —2), and
although the five functions Y im (0, 0) are complex it is
possible to select five real combinations (although it is not
possible to choose five similar shapes). The d j orbital is
concentrated in thezdirection, and the notation z 2 may be
construed as implying a stronger concentration along z than
that of the porbital denoted p The d orbital has its major
concentrations along the bisectors of the x and zaxes, hence
the notation, and so there are four lobes (Fig. H12}. The d 
and the dorbitals are similar in form to d but, as the
X Y xz
notation suggests, are concentrated along the bisectors of the
y and zaxes and the x and yaxes, respectively. The fifth
dorbital is d 2 2, and this peculiar notation implies that
the lobes resemble the lobes of the preceding three orbitals
but are directed along the x and /axes.
2(d) The seven f orbitals, corresponding to S. = 3, and
therefore to a high angular momentum of magnitude (/12)ft,
are the spherical harmonics Y 3m [8, 0), and seven possible
real combinations are illustrated in Fig. H12. It is very un
common to encounter orbitals of higher momentum, but their
form may be deduced quite simply because the angular
dependence of all the "spherical harmonics is known,
3. The radial dependence of the orbitals is contained in the
function R n %{r), and it reveals how closely the electron clusters
around the nucleus. It is of major, but not of sole, importance in
determining the potential energy of the electron and therefore
the energy of the atom. The mathematical form of the
functions had been encountered long before the Schrodinger
equation was solved, for they are essentially the associated
Laguerre functions. Do not be put off by the complexity of
the name: they are simple polynomials in r (Table 14).
All the radial functions tend rapidly towards zero as the
radius becomes large, and this reflects the improbability of
discovering the electron at great distances from the nucleus.
All except the functions R n Ar) vanish at the nucleus, and so
if the electron occupies an o'rbital with S. ¥^ there is zero
probability of discovering the electron actually at the nucleus.
The R n Q (r) functions are peculiar because all are nonzero at
the nucleus, and so all predict a nonvanishing probability of
finding the electron there. Since It = for these functions we
conclude that only for sorbitals is there a nonzero probability
of discovering the electron at the nucleus. The physical basis
of this difference can be understood by recalling that an
electron in an sorbital has zero orbital angular momentum,
and so drifts in toward the nucleus, whereas for all other
orbitals (£ > 0) the angular momentum provides a source of
centrifugal force that flings the electron away from the
nucleus so strongly at short distances that the attractive
Coulombic potential is overcome. Between the nucleus and
infinity the radial wavefunction oscillates a varying number of
times: the number of "nodes in the function R n is n  £ — 1.
The radial behaviour of the functions is depicted in Fig. H13,
and it is helpful to remember that the 1 sorbital {n = !,£ = 0),
which is the lowestenergy orbital, is simply a decaying
exponential function which falls towards zero from a finite
and nonzero value at the nucleus.
4. The complete orbitals are depicted in a few cases in
Fig. H14 and listed in Table 15. The shapes of the orbitals
hydrogen atom
107
FIG. HI 3. Radial vuavef unctions for some states of hydrogen.
Probabilities Inot radialdistribution functions} are shown in colour.
themselves and the probability distributions (which, according
to the Born Interpretation, are the squares of the orbital
functions) have moreorless the same shape, but the
directional dependence is more pronounced. It is inconvenient
to draw such complicated diagrams to illustrate aspects of
atomic and molecular structure and, in accord with the dis
cussion of orbitals, it is common to draw a boundary surface
within which there is a high probability of finding the electron;
this was done to obtain Fig. H 12. Note that the shape of the
surface depends on whether one is attempting to catch a
particular proportion of the amplitude or of the probability.
Beware. The connexion of the boundary surfaces drawn here
should be compared with the depiction of the spherical
harmonics by a similar device on p. 221. The details of the
shape of the orbitals are generally of small importance, and
so boundary surfaces, which are normally drawn with scant
regard for precision, like those in Fig. Ht2, should be inter
preted as rough boundaries for regions of concentrated
amplitude.
5. The most remarkable feature about the structure of the
hydrogen atom is the dependence of the energy on only the
principal quantum number and its independence of the orbital
angular momentum quantum number K. This means that every
state of the atom with principal quantum number n has the
same energy irrespective of the values of C and m^. Since for a
given n the value of £ may range from to n — 1 , and for each
£ the value of m^ may range from — 1 to C, it follows that a
state with quantum number n is/7 2 fold "degenerate. This
peculiar degeneracy, called by the shallow accidental, is a
consequence of the very high symmetry of the central
Coulomb potential, and is lost when the purity of the
potential is destroyed by the presence of the other electrons in
a manyelectron atom (see "penetration and shielding),
6. The energy of the ground state is worth discussing in
greater detail. On classical grounds one expects the electron
to spend as much time as possible in the vicinity of the
nucleus, because by so doing its potential energy is minimized.
The ideal position of the electron is that of contact with the
nucleus. But the ground state of the hydrogen atom, an
electron in a 1sorbital, has an electron distributed in regions
close to the nucleus, but certainly not wholly confined to it.
108
hydrogen atom
hydrogen atom
109
FIG, H14. Amplitude contours of some hydrogen atomic
orbitals.
What repels the electron? The classical answer is its angular
momentum and the concomitant centrifugal force, and this
was the basis of Bohr's model. But with the sorbital there is
associated no angular momentum, and so this cannot be the
explanation. The answer is found in the implication of the
shape of the radial wavef unction and the connexion between
its curvature and the 'kinetic energy.
In Fig, H15 are shown three possible distributions of the
electron in a 1 sorbital. In Fig. H 15c the electron is strongly
confined to the vicinity of the nucleus, but the expense of
doing so is discovered in the sharpness of the curvature of the
wavefu notion: this corresponds to a very high component of
kinetic energy associated with the radial direction. Fig. H15a
shows a situation in which the kinetic energy has been lowered
by permitting the electron a more diffuse domain; but here
the expense arises in the high potential energy of a situation
in which it is allowed to be at a considerable distance from the
nucleus. It is clear that there exists a compromise distribution,
as shown in Fig. H15b, in which the electron achieves a
balance between a moderate kinetic energy and a moderate
potential energy: this is the ground state of the hydrogen
atom. This situation may also be interpreted in terms of the
uncertainty principle (see Questions).
The major features of the structure and spectrum of the
hydrogen atom are well explained by quantum mechanics; but
complications are normally discovered when stones are over
110
hydrogen atom
FIG. H15. Curvature, potential energy, and compromise determining
the ground state of atomic hydrogen,
turned and the ground inspected in more detail. A closer
scrutiny of the spectrum reveals that the lines depicted in
Fig. H10 do in fact have a very fine structure, appropriately
known as »fine structure. This can be explained, as described in
that section, in terms of the coupling of the spin and orbital
momenta, and the energy of the interaction, and the splitting,
emerged naturally as a consequence of the "Dirac equation.
Nevertheless, a slight discrepancy remained even after the
application of that beautiful theory, and the Lamb shift, a
splitting between the levels 2 S % and 2 P^ (which on the Dirac
theory are strictly degenerate), was accounted for only when
the hydrogen atom was treated in terms of quantum electro
dynamics. Nevertheless, the closeness of the predictions of
quantum theory and the experimental results was a triumph of
the theory, and in its turn the explanation of the minute Lamb
shift (a discrepancy of only 0033 cm" 1 } is a triumph of the
more recent modifications of quantum theory.
Questions. 1, What is the evidence concerning the structure of
the hydrogen atom? Estimate the highest frequency transition
in each of the five spectral series. Street lamps containing
sodium often glow red before they turn yellow on account of
the hydrogen they contain: what transition is responsible?
Calculate the "ionization potential of the hydrogen atom from
the frequency of the lines in the Lyman series (Fig. H10). What
is the significance of the quantum numbers n, %, and mn used
to denote an orbital in the hydrogen atom? What is the general
form of the orbitals? What are the features of the sorbitals?
What distinguishes them from all the other orbitals? How
many p, d, f, and gorbitals are there in the case of n = 1, 2,
3, 4? How many nodes does each orbital have? How many
radial nodes, angular nodes, and total nodes does an orbital n,
£, mg possess? What is the connexion between the number of
nodes and the orbital angular momentum? What is the signifi
cance of the number of radial nodes? What is represented by
the curves that are often drawn to denote atomic orbitals?
How many different orbitals correspond to the energy
— /? H /n 2 ? Does the value of mg affect the energy? What is the
physical interpretation of the structure of the hydrogen atom
in its ground state? How would the argument be modified to
account for the structure of the similar species He + ? Discuss
the ground state in terms of the uncertainty principle: con
sider the effect on the "momentum, and therefore on the
"kinetic energy, as the electron is confined more severely to
the vicinity of the nucleus. Discuss the effect on the atomic
orbitals of modifying the nuclear charge: what happens to the
orbitals as the atomic number of the nucleus increases from
1 to 4? What deficiencies are there in the quantummechanical
description of the hydrogen atom?
2. From the Tables (p. 275) plot the radial wavefunction for
the Is, 2s, 2p r and 3dorbitals of the atom. Demonstrate
explicitly that these functions are "normalized. Plot the
probability distribution as a function of r and note that the
most probable position for finding the electron in an sorbital
is at the nucleus. Plot the "radial distribution function for the
Is and 2s orbitals. Calculate the most probable radius for
finding the electron in the Is, 2s, and 2porbitals. Interpret
the different values physically. Calculate the most probable
radius of the atom as a function of Z. At what radius does the
1setectron lie, most probably, in the atoms He, C, F, and U?
Further information. See MQM Chapter 3. For the explicit
solution of the hydrogen atom see Pauling and Wilson
(1935) for the series solution, and Kauzmann (1957), Davydov
(1965), Messiah (1961), and Schiff {1968), For an account by
hyperconjugation
111
the much more elegant method of factorization of the
Schrddinger equation, see Infeld and Hull (1951), Green
(1965), and Englefield (1972). For a discussion of why the
degeneracies are not accidental, see MQM Chapter 3,
Englefield (1972), Bander and Itzykson (1966), Mcintosh
(1959, 1971), and Fock (1935). For further depictions of the
orbitals, and their relation to the classical orbits, see White
(1935). See also Herzberg (1944). For a discussion of the
spectrum of atomic hydrogen, see MQM Chapter 3, King
(1964), Kuhn (1962), Herzberg (1944), and Series (1957).
Through its bibliography the article by Mcintosh (1971) will
set you on a path through abundant fascination: through
accidental degeneracy, projections of hydrogen atoms on to
hyperspheres and hyperhyperbolas, and the four and two
dimensional atom. Theoretical and experimental data for the
hydrogen atom are collected in Table 16.
hyperconjugation. The overfap of oorbitals and jrorbitals
such as the overlap of the methyl C— H obonds with the
aromatic jt orbitals in toluene, is called hyperconjugation, or
the BakerNathan effect. It constitutes a mechanism whereby
the methyl group can behave as an electron donor, and its
consequences include increased electron density at ortho and
para sites, and concomitant effects on aromatic reactivity.
The quantummechanical description considers the three
fjbonds of the methyl group, or, what is equivalent for the
present purpose, the three hydrogen 1sorbitals, as a unit
from which three group orbitals may be constructed
(Fig. H16). If the inphase combination of the three orbitals
is taken, an orbital embracing all three may be formed; this
orbital has cylindrical symmetry about the C— C bond
direction, and so has no net "overlap with the TTsystem. Two
other combinations of the three atomic orbitals may be made,
and both have a single mode. One has its node perpendicular
to the ring, but the other's lies in the plane of the ring, and
therefore has the same symmetry with respect to the C~C
bond as the JTorbital on the ring carbon atom. It can therefore
overlap it, form a very weak bond, and provide a means of
mixing the C— H bond electrons with the TTelectrons.
Further information. See §2.3 of Coulson (1961) and §5.7 of
Streitweiser (1961) for many references. See also Murrell,
plane of ring
FIG. HI 6. Group orbitals on CH3. Only (c) has nonzero net overlap
with the Jrorbital on the neighbouring aromatic ring,
Kettle, and Tedder (1965) and Salem (1966). A discussion
of the evidence and consequences for hyperconjugation are
described by Baker (1952) and in a conference proceedings on
the subject. Baker (1958). Group orbitals are discussed by
McGlynn, Vanquickenborne, Kinoshita, and Carroll (1972),
especially §2.8. Good evidence for hyperconjugation comes
from "electron spin resonance: see Symons (1963), Ayscough
(1967), and Bolton, Carrington, and McLachlan (1962).
hyperf ine interactions. A hyperfine interaction is an inter
action between an electron and a nucleus other than their
pointcharge Coulombic interaction. One may distinguish
between electric and magnetic hyperfine interactions: the
former arises because the nucleus may have an electric
°quadrupole moment, and the latter because it may have a
"magnetic dipole moment. We consider them separately.
Magnetic hyperfine interactions. A nucleus with no n zero
"spin possesses a magnetic dipole moment; this dipole gives
rise to a magnetic field in its vicinity, and with this field the
magnetic moments of the electrons of the molecule may
interact. Consider the important case where the molecule
112
hyperfine interactions
contains a single unpaired electron. This electron possesses a
■magnetic moment by virtue of its spin, and this moment
interacts with the dipolar field of the nucleus by a conven
tional dipoledipole interaction. The interaction energy
depends on the relative orientations of the spin magnetic
moments (just as in the case of two small bar magnets) and
also on the relative disposition of the two spins.
FIG. HI 7. Dipoledipole interaction of parallel moments.
Suppose that there is present an externally applied
magnetic field (as in the electron spin resonance experiment)
which is strong enough to hold parallel the two magnetic
moments of interest. The magnetic interaction energy now has
the classical form for the interaction of two parallel dipoles,
and this depends on/ asr~ 3 and on the angle & as 1— 3cos 2
{Fig. HI 7 and Box Son p. 50}. If the unpaired electron
occupies an sorbital on the magnetic nucleus it is spread
isotropically about it {Fig. H18 a): the net field it experiences
is zero, and so there is no dipoledipole interaction in this
case. If the electron occupies a porbital on the magnetic
nucleus its distribution would not be isotropic, and in fact it
would sample some regions of the nuclear dipole field more
strongly than others (Fig. H18 b). In such a case the energy of
interaction does not vanish.
For a given electron spin orientation (expressed by the value
of the quantum number m ) the energy of interaction depends
on the nuclear spin orientation m r Since a nucleus of spin /
FIG. H18, Field from a nuclear magnetic moment: (a) an selectron
samples positively and negatively directed field equally; (b) a pelectron
samples the positive field more than the negative (in this orientation).
may have 2/ + 1 orientations the electron may experience one
of 2/ + 1 different values of the local magnetic field. In an
"electron spin resonance experiment this leads to a splitting of
the spectrum into 2/ + 1 lines with a separation determined by
the strength of interaction between the two magnetic dipoles.
If the molecule (radical) is rotated relative to the applied
magnetic field, and the moments retain their original
projections m and m., the electron samples different regions
of the nuclear hyperfine field because the porbital distribution
is carried around by the rotating nuclear framework. It follows
that the dipoledipole hyperfine interaction is anisotropic
{depends on orientation of the molecule). If the rotation is
very fast, as it is for a molecule in a liquid, the rotation causes
the electron spin to sample all the different nuclear hyperfine
fields: this spherical average is zero and so the hyperfine struc
ture disappears from the spectrum.
Hyperfine fields at the position of an electron may amount
to tens of gauss. Typical values are given in Table 9. These
hyperfine interactions
113
values are for a pelectron confined completely to the atom
containing the magnetic nucleus; therefore if a smaller value
is measured in an experiment the ratio gives the contribution
of that orbital to a molecular orbital extending over the mole
cule.
Although the above analysis predicts that the magnetic
hyperfine structure of electron spin resonance spectra should
disappear in fluid solution, experimental observation shows
that this is far from the truth, for there usually remains a
splitting of lines which is in many cases considerably larger
than the anisotropic hyperfine structure just described (see,
for instance, Fig. E4 on p, 63). The interaction responsible
for this is the "Fermi contact interaction. It was an approxi
mation in the preceeding discussion that the nucleus gave a
truly dipolar field. That would be sufficiently accurate if the
nucleus were vanishingly small or if an electron could never
approach it sufficiently closely to realize that it was not an
infinitesimal point. The nucleus does have an extension (its
diameter is of the order of 10" 14 m) and if the electron is in an
sorbital it may approach the nucleus very closely. When it
gets there it discovers that the field is not a pure dipolar field
(see Fig, F1 on p. 72). Let us take the view that the electron
actually penetrates the nucleus, and let us replace the nuclear
magnetic moment by an equivalent current loop with the same
radius as the nucleus. Outside the loop the field does
average to zero when the electron spin samples it spherically;
but inside the field is all in the same direction, no cancellation
occurs, and the spin magnetic moment interacts with the non
vanishing average of the field. This interaction is isotropic
because it is independent of the orientation of the remainder
of the nuclear framework. It is unique to selectrons because
all others have a node running through the nucleus and so
cannot penetrate it.
The Fermi contact interaction may be very large: some
typical magnetic fields experienced by the electron by this
mechanism are also listed in Table 9. Since it is characteristic
of sorbital character the magnitude of the observed isotropic
splitting may be used to determine the amount of selectron
character in a molecular orbital.
Now let us turn off the external magnetic field, and let all
the competition for magnetic coupling be between interactions
within the molecule. The nuclear magnetic moment may
couple to the magnetic fields from various sources within the
molecule, and the energy of this interaction will appear in the
spectrum (particularly the rotational spectrum) as a splitting
of lines; molecules with different nuclear spin states will give
rise to slightly shifted spectra, and therefore the observed
spectrum will be the collection of lines from all the different
molecules in the sample. The other sources of magnetic
moment in the molecule include the electron spin (which now
couples differently because there is no applied field holding it
parallel to some laboratory axis), the electron orbital
magnetic moment, and the 'molecular magnetic moment.
Electric hyperfine interactions. It is not possible for a
nucleus to have an electric dipole moment (on grounds of
symmetry), but it may have an electric 'quadrupole moment
if its spin quantum number / is 1 or larger. The presence of an
electric quadrupole moment implies an asymmetry in the dis
tribution of charge in the molecule, and there are two possi
bilities. These are illustrated in Fig. H19. In one, the prolate
case, there is an excess of positive charge in the polar regions
of the nucleus (with respect to the axis of spin), and a com
pensating slight relatively negative band in the equatorial
zone. In the oblate case the distributions are reversed. An
electric "quadrupole moment interacts not with an electric
field itself but with the field gradient. If, therefore, there is an
electric field gradient at the quadrupolar nucleus, the energy
of the molecule will depend on the orientation of the nucleur
in the molecular framework, and so we arrive at another
hyperfine interaction.
distribution
of pcwitive choge
©
low eneigjr high enenjy
FIG. H 19. A nuclear eluctric quad ru pole in a field gradient.
114
hyperfine interactions
Is there a field gradient at a nucleus in a molecule? If the
surrounding electrons are in an sorbital, or constitute a
closed shell, there is no field gradient. If however the electrons
do not form a closed shell and are not selectrons there may be
a very strong field gradient (see Fig. H19 and Fig. Q2 on
p. 187). Therefore the magnitude of the electric quadrupole
interaction in a spectrum gives information about the state of
"hybridization of the orbitals occupied by the electrons sur
rounding it (that is, the amount of porbital character in the
surrounding electrons). There are, of course, complications.
One especially important nuisance is that the valence electrons
may distort the underlying closed shells. This distortion gives
rise to a field gradient at the nucleus and therefore increases or
decreases the strength of the quadrupole coupling: these are
the Sternheimer antishielding and shielding effects. The dis
tribution of charges in the vicinity of the nucleus may also be
estimated by pretending that the molecule consists of an array
of point charges (see dipole moment): ionic it y has been
estimated in this fashion.
Questions. What is meant by the term 'hyperfine interaction'?
What two classes of interaction are there? What is the nature
of the magnetic coupling between an electron spin and a
nuclear spin? Why does this interaction disappear (when there
is an external field present} when the electron is distributed in
an sorbital, or when it is in a porbttal but the molecule is
rotating rapidly in a liquid? What information can be obtained
from the magnitude of the anisotropic hyperfine interaction?
Why does there remain an interaction in fluid solution? Of
what is the Fermi contact interaction diagnostic? In the
radical N0 2 the hyperfine interaction to the nitrogen nucleus
is 55 G for the isotropic component and 13 G for the
maximum of the anisotropic component: what is the s and p
character of the nitrogen orbital contributing to the molecular
orbital occupied by the unpaired electron? (Use Table 9.)
What is the source of the electric hyperfine interaction? What
nuclei may possess a quadrupole moment, and what does such
possession signify? With what electronic property does the
quadrupole interact? What information may be obtained from
a study of the interaction? What is the complicating feature?
Further information. See MQM Chapter 1 1 for a more detailed
account of the magnetic hyperfine interactions. The way that
they are employed in "electron spin resonance and "nuclear
magnetic resonance (especially for spinspin coupling
constants) is described by McLauchlan in Magnetic resonance
(OCS 1). See also Lyndenfiell and Harris (1969), Carrington
and McLachlan (1967), Atkins and Symons (1967), Ayscough
(1967), Bolton and Wertz (1972), Slichter (1963), and
Abragam and Bleaney (1970). The role of hyperfine inter
actions in spectra, and the structural information they may
be used to obtain, are described in Sugden and Kenney
(1965), Herzberg (1950), and Townes and Schawl ow
(1955). See Freeman and Frankel (1967) for a basic dis
cussion. Magnetic and electric properties of nuclei are listed
in Table 17.
I
ionization potential. The ionization potential (IP) is the en
ergy required to remove to infinity an electron from the orbital it
occupies in an atom or molecule. The more tightty bound the
electron the greater its ionization potential. The energy is
normally expressed in electronvolts (1 eV is equivalent to
96'49 kj mof 1 ), and its value depends on the orbital involved
and the state of ionization of the atom. The first IP is the
energy required to remove the least tightly bound electron
from the neutral atom, the second IP is the energy required to
remove the least tightly bound electron from the singly
charged ion, and so on. IPs are a good guide to the chemical
properties of elements, and the periodic table can be discussed
in their light. A useful exercise of this nature is the discussion
of the IP of the atoms from He to Ne along the first row of
the periodic table (see Table 18). These can be rationalized in
terms of "penetration and shielding effects.
Questions. What is an ionization potential? The ionization
potential of atomic sodium is 514 eV; how much heat is
evolved when 1 mol of sodium atoms is formed from a gas of
Na ions and electrons? Could an atom have a negative
ionization potential? Account for the IPs of the elements of
the first row in terms of penetration and shielding effects.
Further information. Lists of ionization potentials will be
found in Kaye and Laby (1956) and § 7b of Gray (1972). In
Puddephatt's The periodic table of the elements (OCS 3) will
be found a discussion of their dependence on penetration and
shielding effects, and the role they play in determining
chemical properties. This aspect is developed in detail by
Phillips and Williams (1965). The ionization potential is used
to calculate "electronegativities, and is a basis of the experi
mental technique of photoelectron spectroscopy (see
"photoelectric effect).
intermolecular forces: a Synopsis. The principal classifi
cation of the forces between molecules is according to their
range. The shortrange forces are repulsive, and reflect the
increase in energy that occurs when two electron clouds are
forced into contact and penetration. As two closedshell
species approach each other some of the electrons may be able
to adjust their distribution so that they occupy energetically
favourable regions, but the "Pauli principle does not permit all
the electrons to settle into these regions, and the remainder are
forced to occupy regions that tend to raise the energy of the
approaching pair (they enter the "antibonding orbitals in the
"molecularorbital theory). This disruptive effect overcomes
the attractive effect of the more favourably sited electrons,
and so the energy of the pair of species rises sharply as the
distance diminishes: this may be interpreted as a force that
drives the molecules apart, and so we see the repulsive force to
be rooted in the operation of the Pauli principle.
Longrange attractive forces between molecules must exist,
for otherwise no condensed phases would exist. The simplest
attractive interactions are between charged species (ionion
interactions, the tonic "bond), where the Coulomb force draws
the components together until repulsive forces supervene.
Most molecules are uncharged, but attractive forces still
operate. In polar molecules these can be identified with the
attractive interaction between the permanent electric moments
of the molecules (the 'dipoledipole interaction, or the dipole—
115
116
intermolecular forces: a synopsis
( » R < a
\ R > 0,
BOX 10: Intermolecular potentials
Rigid spheres:
VI
Point centres:
ViR) = DR 6
9 < 8 < 15 usually. Maxwell tan molecules have 5 = 4.
Square well:
!°° R < O
e o < R < ra
R > ra.
Sutherland:
v *Hcm r>o.
LennardJones:
Vift\ = DR~*  CR^
special case: (6, 12)potentia1
Buckingham :
V{R) = 5e _dfl  CR^  C'R~ % .
Modified Buckingham (6exp):
[e/(16/a]] [ (6/a)exp(oa/?/ff )R*JR 6 1
R <R ™
R is the value of R for which the upper expression for
m
ViR) attains a maximum.
Stockmayer:
V[R) = V LJ iR) + S.iR,Huth).
where V AR) is the Lennard Jones potential, and
£(R, fi lr fa) is the dipoledipole interaction energy given in
Box 5.
ViR)
{'
pointcharge interaction if one is charged, or the dipole—
induceddipole interaction if one is polar and the other not: in
the last case the strength of interaction depends on the
"polarizability of the nonpolar molecule). Then there are the
forces between nonpolar molecules: the most important of
these is the London "dispersion force which isal so termed an
induceddipole— induceddipole interaction because it depends
on the fluctuation of the electron density of one molecule
leading to an instantaneous dipole moment which may in turn
induce an instantaneous dipole moment in the other, these
two dipoles then sticking together. This interaction depends on
the polarizability of both molecules and its energy depends on
their separation as R' 6 .
The name van der Waals forces is a general term applied to
these intermolecular interactions. The term Keesom force is
reserved for the interaction between polar molecules, and
London force refers specifically to the dispersion force.
The distancedependence of the forces and the angular
dependence of those between nonspherical species are
normally expressed in terms of an empirical formula which
has moreorless the correct qualitative form and contains only
a few adjustable parameters. Some of the more common inter
molecular potentials are illustrated in Fig. 11 and Box 10.
All make a crude approximation to the repulsive interaction
which, in reality, because it depends on details of molecular
■wavef unctions and overlap, has a complicated behaviour.
The most cavalier approximation is to replace the repulsive
part by an impenetrable hard sphere. The LennardJones
potential assumes that the potential rises sharply at small
separations according to R' n ; when, for mathematical con
venience, n is chosen to be 1 2, we have the {6, 1 2}Lennard
Jones potential (the R~ 6 dependence [see Box 10] reflectsthe
presence of the dispersion forces). The Buckingham exp6
potential retains the R' 6 component but pretends that the
repulsive forces vary exponentially. The Keesom potential is
an expression for the interaction of nonspherical molecules
which treats them as cylinders capped by hemispheres. The
Stockmayer potential adds to the LennardJones potential a
term representing the interaction of two dipoles placed at the
centres of the molecules.
Intermolecular forces must be taken into account if the
properties of real gases and liquids are to be understood, and if
inversion doubling
117
rigid spheres
poinf centres
_o;
Sutherland
Buckingham
FIG, II. The shape of some empirical iniermolecular potentials.
quantitative calculations are to be made. One important route
to thermodynamic quantities is through the virial equation of
state, where PV is expressed as an expansion in 1/1/; the coef
ficents in this expansion, the virial expansion, are the virial
coefficients. Expressions for these may be obtained from the
intermolecular potentials and the coefficients used to calculate
the properties of the gas. The structure of liquids depends on
the form of the intermolecular forces. The forces {and the
parameters in the empirical expressions) may be determined
by fitting calculated virial coefficients to experimentally
determined values, from transport properties, and, best of all,
from scattering experiments In molecular beams, where
individual molecules are used both as targets and projectiles
and their deflexion is determined by their interaction.
Further information. An excellent book on the subject, which
has a synoptic review in an early chapter and detailed accounts
in later chapters, is that by H irsc hf elder, Curtiss, and Bird
(1954). A helpful account of Debye's contribution has been
given by Chu (1967), and details of the quantummechanical
equations involved, and their derivation, will be found in
MQM Chapter 11, Kauzmann (1957), and Margenau and
Kestner (1969). The determination of intermolecular poten
tials is described by Hirschf elder, Curtiss, and Bird (1954),
Curtiss (1967), and Dymond and Smith (1969). Molecular
beams and their application to the determination of inter
molecular potentials are described by Levtne and Bernstein
{1974) and Ramsey {1956}, and reviewed in a collection of
articles edited by Ross (1966). Experimental and theoretical
aspects of intermolecular forces have also been reviewed by
Hirschfelder (1967).
inversion doubling. Inversion "doubling can be discussed
in terms of the specific and important example of the
ammonia molecule. This pyramidal molecule can vibrate in a
symmetrical bending mode, rather like an umbrella being
shaken dry. And, like an umbrella, there is some probability
that it can be inverted into another configuration where it
continues to vibrate. The original and inverted configurations
are physically indistinguishable, and vibrate with the same
frequency; this situation can be pictured as in Fig. 12, where
the molecule vibrates in one of the two potential wells. But
118
inversion doubling
frequency
FIG. 12. Inversion doubling in ammonia: black lines are the 'confined'
states of the two wells; coloured tines are the true, interacting states.
the fact that the molecule has some chance of inverting, or
of "tunnelling from one well into the other, implies that the
"vibrational wavefunctions of one well (which represent the
displacement of the nuclei) seep through into the other. If
the vibrational wavefunctions were wholly confined to one
well the molecule would not be able to invert. Since the
wavefunctions seep through the potential barrier their
amplitudes overlap, and, just as in the case of the formation
of a molecular orbital, the correct description of the
vibrational state of the molecule is in terms of a super
position of the functions. Thus we take either their sum or
their difference. The two new functions so formed correspond
to different energies, because one is concentrated more with
in the barrier than the other. It is important not to conclude
that the lower state is the one with the node within the
barrier, even though this has the lower potential energy. We
must also, as always, take into account the kinetic energy,
and discuss the total energy. When a node is introduced into
a function it becomes more curved, and its kinetic energy
rises. This is the case for the tunnelling states of ammonia, and
an analysis shows that the kinetic energy dominates and that
the unbuckled, nodeless, symmetrical function corresponds
to the lower total energy. In this connexion see torsional
barrier. It follows that, instead of having pairs of "degenerate
vibrational states in the ammonia molecule, the inversion
causes the degeneracy to be removed and each pair splits into
two: each level is "doubled.
The energy separation depends on the difference in
amplitude (and therefore probability) that the nuclei will be
found within the potential barrier: this depends on the
strength of the overlap between the vibrational wavefunctions
in the two wells, and this in turn depends both on the height
of the potential barrier above the interacting levels (the higher
it is the smaller the penetration) and on the mass of the
"tunnelling species (the greater the mass the less the tunnel
ling). In NH 3 , the splitting, which is known as the inversion
frequency, is 23 786 MHz (0793 cm 1 ) for the lowest level.
On a classical picture of the process this frequency would be
identified with the frequency with which the inversion
actually occurred. In quantum mechanics that concept is
untenable in detail, but a "wave packet localized in one well
(and representing one configuration of the molecule) would
wriggle through the barrier and emerge on the other side, with
more or less the observed 'inversion frequency'.
Questions. 1. Why is inversion doubling so called? What is its
source? Why do the positive and negative combinations of the
vibrational levels correspond to different energies? Which has
the lower energy? What determines the separation? Discuss the
dependence of the separation as the barrier height is reduced
from infinity to zero. How does inversion doubling affect the
spectrum of ammonia? What changes would you expect to
observe on deuteration of ammonia? Discuss inversion
doubling in the other Group V hydrides.
2. Consider a double squarewell potential with a rectangular
potential barrier; take the potential V to be infinite for
x<0and* >£, to have the values between x= ^L — b and
jt +6, and to be zeroelsewhere.Set up and discuss the solutions
of the "Schrb'dinger equation for a particle of mass m in this
system, and discuss the doubling of the energy levels that is
brought about by lowering the height of the barrier from
infinity. Discuss the effect of changing the mass of the
particle.
Further information . See MQM Chapter 10. For the spectro
scopic consequences of inversion doubling see §9.19 of King
inversion doubling
119
(1964), §8.2 of Sugden and Kenney (1965), various parts of
Herzberg (1966), and Chapter 12 of Townes and Schawlow
(1955). The inversion process is the basis of maser action (a
microwave iaser process) and so has been studied in consider
able detail: for applications of this nature see the references
under "laser, Vuylsteke (1960), and Troup (1963).
J
J. The letter J is worked hard in quantum theory, but the con
text normally eliminates confusion. J and/ are 'quantum
numbers used to denote the total "angular momentum of a
system: the former is used in a manyelectron system, or for
the overall "rotation of motecufes, and the latter is used when
only a single particle is involved. As a consequence of this use
both J and/ are used to distinguish the "levels of 'terms. J is
also used to denote the ^spinspin coupling constant in °n.m.r.
and the 'Coulomb integral for the electrostatic interaction of
electrons. J or j is used for the current, be it the current of
electrons induced by applied fields in metals, atoms, and
molecules, or the flow of matter, heat, and entropy.
JahnTelter effect. To those for whom the natural tendency
of Nature is to states of highest symmetry, the JahnTeller
theorem is a bitter pill, for itscontention is that in a variety of
situations a molecule of high symmetry is intrinsically unstable
and will attain a lower energy by distorting into a lower
symmetry configuration. To be more precise, the theorem
states that no nonlinear molecule can be stable in a
"degenerate electronic state. Therefore, if the molecule is
found, by calculation, to be degenerate, that form of the
molecule will not be the stable form in Nature, and a less
symmetrical and nondegenerate form will be its natural state.
The theorem does not apply to linear molecules, and so these
may exist in an undistorted degenerate state; but there is an
analogous higherorder effect, the 'RennerTeller effect, which
takes care of that loophole.
As an example of the JahnTeller effect we shall consider
the case of a Cu 2+ ion in an octahedral environment. We
proceed to deduce its electronic structure on the basis of
crystal fie Id theory (which is good enough for our present
purpose), and discover that its configuration ist 6 e 3 . This is a
degenerate configuration because the two configurations
d 2d 2 2 and d 2 2d 2 have the same energy. The
2 x v x y z 3 '
molecule is nonlinear, and so the JahnTeller theorem
predicts that the molecule must distort and eliminate its
degeneracy.
The physical reason for this may be understood by con
sidering the forces operating on the iigands: in the first
configuration there is more electron density along thezaxis
than in the equatorial plane, whereas the opposite is true in
the second configuration. In the first case there is a tendency
for the Iigands along the zaxis to move away from the central
ion, and for the equatorial Iigands to move in; in the second
configuration the opposite shifts are to be expected. If the
molecule does distort by stretching along the zaxis, the
electronic configuration d* 3 d x 2_ 2 will have a lower energy
than the other electronic configuration in the same environ
ment {Fig. J1 a). If the complex flattens along the £axis, for
that conformation the second configuration will have the
lower energy (Fig. J T b). Therefore we see that there does
indeed exist a distortion of the molecule that removes the
degeneracy of the electronic states, and we should expect the
molecule to be found in either the elongated or the compressed
form. (Which form it attains is very difficult to predict.)
Several electronic configurations are predicted to produce
JahnTeller distortions in octahedral complexes: we simply
have to seek cases in which degenerate configurations can arise.
120
JahnTeller effect
121
FIG, J1. The effect of (a] elongation and (bl flattening an octahedron
on the energy of electrons in d 2 and d 2 2.
z x —y
The highspin d 4 configurations t 3 e l , the lowspin d 7 con
figurations tV, and the d 9 configurations tV are all cases in
which there is degeneracy because a single electron or a single
hole has to occupy the two degenerate eorbitals (d % and
^22) Examples are Cu 2+ (d 9 ) and Cr 2 * (d*), whose com
pounds often have closely related distorted structures, Mn J+
(d 4 ), and the rare lowspin Co 2+ (d 7 ), lowspin Ni 3+ (d 7 ) r and
Ag 2+ (d 9 ). JahnTeller distortions are not expected for d 3 ,
highspin d 5 , lowspin d* , or d 8 .
JahnTeller distortions might be predicted also for con
figurations that give rise to degeneracy in the torbitals, such
as d l , d 1 , highspin d 6 , and highspin d 7 . In practice the
torbitals are directed between (rather than along) the metal
ligand axes, and uneven electron occupation therefore fails
to produce observable distortions.
The d 8 configuration is rather special in the sense
that small distortions are not expected; if a distortion does
occur it is large. Because the spins are parallel, one electron
must occupy d 2 and the other 1 must occupy d 2 2 ;there
z x — y
fore in a small distortion the energy of one electron increases,
the other falls, and so overall there is no change in the energy.
If the distortion is great enough, however, the energy differ
ence between the d 2 and d 2 2 orbitals can exceed the
energy required to cram both electrons into either d 2 or
d i 2 (that is, to overcome the first Hund rule). Strictly
x — y
speaking, this is not a JahnTeller distortion because it does
not lower the degeneracy of the system.
Two effects complicate the analysis of the effect. The first
is the dynamic JahnTeller effect, in which the centre of
attention is the motion of a molecule in which a degeneracy
may arise at certain nuclear conformations. The second is the
role of °spinorbit coupling, which generally reduces the
effect's magnitude. The experimental detection of a true Jahn
Teller distortion is very difficult as there are other reasons why
a complex may be distorted. In particular it is difficult to
distinguish it from a distortion due to the packing requirements
on formation of a crystal.
Questions. What is the JahnTeller theorem? What molecules
are excluded? When a molecule is predicted by calculation to
have a degenerate electronic state, what are the consequences
of the theorem? Account for the content of the theorem in
terms of the crystalfield model of transit ionmeta! complexes:
why may a lower energy be attained when an octahedral com
plex is distorted? What numbers of electrons must such a com
plex possess in order for this description to apply? Why is the
JahnTeller effect much less important in tetrahedra!
complexes? What situation arises when the configuration of
the central ion is d 8 ? What are the methods for distinguishing
the JahnTeller effect, and what are the complications? When
is the dynamic JahnTeller effect important? What is the
'ligandfield (MO) explanation of the JahnTeller effect?
Further information. See MQM Chapter 10. A simple account
of the JahnTeller effect is given in §4.2 of Orgel (1960) and
Coulson (1961). Some of the mathematics is put in a simple
way in §8d of Ballhausen (1962), and a full and interesting
discussion of the spectroscopic consequences is given in §1.2
of Herzberg (1966). A discussion of both the static and
dynamic effects is given in a book devoted to the subject by
Engl man (1972), and a simple example is worked on
p. 45 of Herzberg (1966) and on p. 194 of Ballhausen (1962).
K
kinetic energy. The kinetic energy of a particle is the energy
it possesses by virtue of its motion. In quantum mechanics the
kinetic energy is related to the curvature of the wavefunction.
As the wavefunction becomes more sharply curved, so the
kinetic energy of the state it represents becomes greater. Con
versely, a state represented by an almost flat function has
virtually no kinetic energy.
In the case of a free particle the wavefunction is of infinite
extent and has a wavelength that decreases as the kinetic
energy increases. The reason for this can be grasped quite
easily from the connexion between curvature and energy: as
the curvature of the wavefunction increases it becomes more
buckled, and swings more rapidly from positive amplitude to
negative; but increasing the rapidity with which the function
changes sign is simply another way of saying the wavelength
shortens. This connexion of wavelength and kinetiG energy,
and thence momentum, of free particles is the explanation of
the de Broglie relation.
In cases where the particfe is bound to a potential centre the
wave may not be sufficiently extensive for a wavelength to be
discernible or meaningful, but the relation between curvature
and kinetic energy remains. An example of this situation is the
"hydrogen atom: the ground state arises from the balance of
the kinetic and potential energies; the 1sorbital is a simple
exponentially decaying function which never passes through
zero, yet an electron in it possesses kinetic energy by virtue of
the nonzero mean curvature of the exponential function.
Questions. 1. What properties of the wavefunction determine
the kinetic energy of a system? What is the connexion between
the wavelength of a free particle and its kinetic energy? How
is the wavelength related to the momentum? Calculate the
wavelength of an electron with an energy equivalent to 1 eV,
1 keV, 1 MeV {eV is "electronvolt). Discuss the role that the
curvature of the wavefunction plays in determining the
structure of the ground state of the hydrogen atom; what
would happen to the energy of the atom if the electron were
pinched more closely towards the vicinity of the nucleus in
order to lower its potential energy?
2. The kinetic energy of a onedimensional system is calculated
by evaluating the expectation value of the "operator— {b 2 /2rn)
(d /dx }, and the corresponding operator for a three dimensional
system is {h 2 /2m) {9 2 /3x 2 + d 2 /V + d 2 /dz 2 ), or
—{\i 2 /2m)V 2 in terms of the "laplacian. Evaluate the kinetic
energy of a particle with the following wavef unctions:
explkx, s'mkx, stnnkx s\r\mky, exp( kx 2 ), exp(— nr/a ).
Plot the kinetic energy of a particle with the last wavefunction
as a function of n, and so see the connexion between the
curvature of a hydrogenlike exponential function and the
kinetic energy.
Further information. See MQM Chapters 3 and 4 for some
more specific examples. All standard books on quantum
theory bring out the connexion of the curvature and the
kinetic energy: therefore see Chapter 5 of Davydov (1965),
Landau and Lifshitz U958a), and Messiah {1961). An
interesting account that emphasizes the opticalmechanical
analogy is given in Chapter 3 of Bohm (1951), Scattering
phenomena are excellent examples of the wavelength's
connexion with kinetic energy; therefore see books on
122
Koopmans' theorem
123
scattering theory, such as Rodberg and Thaler {1967) and
Goldberger and Watson (1964), as well as the appropriate
sections in the other books referred to above. Levine (1969)
applies scattering theory to the discussion of molecular
reactions; and this is presented in a more pictorial form by
Levine and Bernstein (1974),
Koopmans' theorem. 'Theorem' is a name too grand for
this approximation; Koopmans' rule would be better, for it is
transgressed frequently, and is never obeyed to the letter. The
rule states that the 'ionization potential of an electron is equal
to the energy of the orbital from whence it came. This appar
ently trivial statement is based on the fact that the energy that
has to be supplied in order to ionize an atom or molecule must
be expended in overcoming the combined effect of nuclear
attraction energy and the electron's repulsive interaction with
the other electrons present, and these factors determine its
energy in its orbital.
The rule is an approximation because it assumes that the
remaining electrons will not reorganize themselves in order to
take advantage of the absence of the electron which is being
removed. Therefore, when oneelectron energies are calculated
in the HartreeFock "selfconsistent field manner, the ion
ization potentials calculated by Koopmans' rule are often in
error because it is assumed that the electrons in the ion occupy
the same orbitals as they did in the atom. Furthermore, the
HartreeFock scheme neglects electron "correlation effects,
and is nonrelativistic: the latter approximation can be
seriously in error for electrons that are strongly bound and so
subject to strong forces.
Questions. 1. State Koopmans' rule. What is its justification?
Why is it only an approximation? Would you expect it to
overestimate or underestimate ionization potentials? For
which electrons should it be a better approximation? For
what species is the rule exact?
2. The oneelectron energies of the CO molecule are as follows:
4a, 21 '87 eV; 50, 15 09 eV; 17T, 1740 eV. Estimate the ion
ization potential for the molecule when an electron is removed
from these orbitals (experimental values are 1972 eV, 1401
eV, 1691 eV). Discuss the structure of the molecule in the
light of these values (which are taken from p. 36 of Richards
and Horsley, loc. cit. infra).
Further information. Koopmans' rule is now of interest
because of the development of photoelectron spectroscopy:
for an account of this see Turner, Baker, Baker, and Brundle
(1970), Molecular energies are listed in the bibliography com
piled by Richards and Horsley (1970).
L
laplacian. Pierre Simon de Laplace (17491827) was a
notable French mathematician, and to his work his formidable
Mecanique celeste is a profound and worthy memorial.
Among its many notable pages of mathematics is the Laplace
equation, ® 2 flbx 2 \ + (b 2 f/by 2 ) + (b 2 f/bz 2 ) = 0, where f is
some function. Although the equation was set up in order to
account for the properties of gravitational fields it has turned
out to be applicable to a wide variety of phenomena. That it
governs properties like the flow of incompressible fluids,
BOX 11: The laplacian and the legendrian
Cartesian form (x, y, z as coordinates}
laplacian: *& + £+£.
Spherical polar form {r, 8, as coordinates)
laplacian; V 2 =
t(&)^
A?
or V'
2 0_
r br
_ + i. 2. + J_ a 2
b , /; 9
0b6 smu ae
or A :
sinO 90 ' 30 2
. cosfJ b_ ,
Cylindrical polar form (r, Q, z as coordinates)
laplacian: V 2 = f +
+ ^
1 d 2
r or br 2 r 2 b6 2 bz 2
gravitational and electromagnetic fields, and heat flow is
probably due to the fact that it is an equation that expresses
the tendency of natural phenomena towards uniformity: to
the elimination of curvature in the distributions. So important
is it that the differential operator {d 2 ibx 2 ) + (3 2 /by 2 ) +
(b 2 !bz 2 ) is given a special symbol V 2 the laplacian, and read
'del squared'. Laplace's equation then becomes simply v*V= 0.
It is not always convenient, and often foolish, to work with
the laplacian expressed in cartesian coordinates; for systems of
a predominantly spherical nature the spherical polar form is
more convenient, and both forms are collected in Box 11. The
part of the laplacian involving the angular derivatives is referred
to as the legendrian and written A 2 .
Further information . The laplacian occurs widely in quantum
mechanics because it is the "operator for the "kinetic energy;
it therefore appears as a component of the "hamiltonian and is
the differential part of the °Schrddinger equation. For an
account of its transformation from cartesian coordinates to
spherical polars see §6.8 of Kyrala (1967) for swift methods
and Appendix 5 of MoelwynHughes (1961) for slow. A brief
biography of Laplace has been written by Newman (1954).
laser. The word laser is an acronym formed from 'light
amplification by stimulated emission of radiation' and is a
development of the maser, where the m denotes microwave,
according to some, or molecular according to others. A laser
operates by absorbing energy and emitting it at a welldefined
wavelength by a stimulated emission process (see "Einstein A
and fi coefficients). As a simple example, consider a sample of
124
material in which most of the atoms are in an excited state,
and let the sample be contained in a cavity with reflecting
walls. One atom will emit a photon as it falls spontaneously
into the ground state, and this photon will rattle around inside
the cavity, its presence stimulates another atom to emit, and
so a second photon appears in the cavity and travels in phase
with the first. The pair of photons stimulates another
emission, and very rapidly a cascade of emission occurs and an
intense light field grows inside the cavity. The process ceases
when the population of the excited state has fallen to its
equilibrium value. If one of the walls of the cavity is semi
transparent, light will leak through it and an intense, mono
chromatic burst of phasecoherent light will emerge.
How is the process realized in practice? The cavity is
normally in the form of a long thin tube, if the sample is a gas,
or a cylinder of material if it is a crystalline solid; this configur
ation implies that the emitted radiation will have a very small
lateral divergence because only those photons ricocheting
backwards and forwards along or very close to the axis con
tribute to the amplification cascade: the remainder pass
through the side walls and drain away from the system before
their intensity amplifies. Thus the geometry of the system
leads to a beam with very little divergence.
The principal problem that remains is how the population
inversion necessary for stimulated emission, and therefore for
IB
purrp
c=&
light
pumpCr^
IITW1 1
JiaSt
heat
liWlT
'radiotionless
I decay
FIG. L1. (a) A iwoleve! laser and (b) a threelevel laser.
125
laser action, is achieved. Two basic systems may be envisaged.
In the first (Fig. L1 a) the laser action takes place between an
excited state and the ground state: the difficulty of this
primitive arrangement is that it is necessary to remove more
than 50 per cent of the atoms from their ground state into the
excited state, The second method obtains laser action between
two excited levels: the populationinversion is much easier to
attain, especially if the lower excited state can relax rapidly
into the ground state. In such a threelevel laser {see Fig. LI b)
intense radiation excites the absorbing atoms or ions into the
uppermost level, and if this pumping step is sufficiently
effective the population in that level {E 2 ) is significantly, and
even enormously, greater than that in the lower excited level
E t . If the E 2 population does not leak away into the lower
levels by nonradiative processes, laser action may take place
between E 2 and E , because photons passing back and forth
within a reflecting cavity stimulate the B%~ *fi emission; a
leaky mirror at one end of the cavity permits a highly
collimated {nondivergent), coherent (all waves in phase,
because they are generated by stimulation), polarized (because
of the polarization of the transition or because the cavity has
polarizing windows}, and monochromatic radiation of
frequency P= (f 2  E h )lh to emerge.
An example of the laser process is the heliumneon laser in
which the active material is a mixture of the two gases in the
ratio 1 :5. A radio discharge, being no respecter of "selection
rules, excites the helium atom, and although many of the
excited states decay very rapidly, the first excited singlet state
is relatively longlived (because ss transitions are forbidden),
and atoms in this state swim around in the sample for some
time because they cannot return to the ground state except by
a forbidden transition. Whilst swimming an excited helium
atom may collide with a neon atom in its ground state, and by
a coincidence (a coincidence at the heart of this laser system)
the energy that the helium atom possesses almost exactly
matches the energy of an excited state of neon. There occurs
resonant transfer of energy, the helium is deactivated and the
neon is in an excited state E 3 (Fig. L2). This state has two
unpopulated energy states beiow it which do not resonate
with excited helium. Therefore it is in a configuration typical
of a threelevel system; laser action occurs between E 3 and E\,
and red light is emitted at 632'82 nm.
126
laser
He
r
2'S
4^y === ^
2h
discrieitje ■ — /
ft.
collision
FIG, L2. The Me— Ne laser (for the 6328 nm mode).
Ruby is a famous laser, and is an example of a sort of three
level system involving four levels, but acting like a twolevel
system in so far as laser action involves a transition to the
ground state. These somewhat perplexing remarks can be
resolved by a glance at Fig. L3. Ruby, which the prosaic
know to be Al 2 3 with about 5 per cent of the Al 3+ ions
heat
heat
pump
6943 .....
FIG. L3. The ruby laser transitions.
replaced by Cr 3 *, gives delight to poets by virtue of its two
intense transitions in the green (to E 2 ) and the violet (to E 3 ).
Having pleased poets, ruby proceeds to please physicists as its
chromium ions drop from E 3 and E t into E t : this is a
radiation/ess decay and relies on the transfer of the electronic
excitation energy into the vibrations of the surrounding lattice
(which gets hot). The initial absorption is so efficient that
when intense illumination is used the population of E t may
exceed that of the ground state, and laser action occurs with
the emission of red light {694'3 nm).
Lasers may be either continuous wave (CW), when the light
emerges in a continuous flow so long as the pumping
operation is in progress, or pulsed, as in the case of ruby, when
the laser light emerges in a short burst as the stimulated
emission occurs. The emergence of the light in short bursts
means that very high powers may be obtained, albeit only for
very short times. A typical procedure in a ruby laser is to
employ a brilliant pumping flash emitting about 25 kJ of
energy. Much of this is absorbed by the ruby rod, and although
much of the absorbed energy appears as heat, about 25 J
appears as laser radiation. But the pulse of light lasts for only
about 5 X 1 CT's, and so the power that emerges is a splendid
50 kW, but of course this power is maintained for only
5X10 s. It is possible to increase the power output by
shortening the length of the pulse in which the same amount
of energy is delivered: the technique of effecting this is
Qswitching. A crude way of describing this is to imagine a
cavity with one of the mirrors removed; then pumping
radiation is applied in a flash and the population of the upper
level attains its maximum value and is not depleted by
stimulated emission. Then the lost mirror is hurriedly slapped
back into place while the population is still inverted, and it
all drops out, by stimulation, and a giant pulse is obtained in
about 10~ 8 s. The power of the laser, if the pulse carries 25 J,
is a massive 25 X 10 9 W, which is roughly the output of a
massive power station (but power stations have the
economical advantage of working for some years at that level,
rather than for 10~ 8 s,. The removal and replacing of the
mirrors was done mechanically in early models but now the
Kerr cell or Pockels cell is an electrical substitute.
Some of the common laser materials and their wavelengths
are listed in Table 19.
127
Further information. See MQM Chapter 10 for a further dis
cussion of the ruby system, A simple account of lasers is given
by Schawiow (1969), Wayne (1970), and Lengyet (1971), and
in review articles by Jones (1969) and Haught (1968). The last
gives information about the expressions modelocking (a
manner of achieving short, picosecond flashes), Qswitching,
various laser systems, and many pertinent references. A simple
account of liquid lasers has also been given by Heller (1967).
The stimulating source need not be radiation: chemical re
actions that leave product molecules in excited states are also
the basis of laser action— a chemical laser is a remarkable
device that turns chemical energy directly into coherent light.
For an account see Haught (1968) and Levine and Bernstein
(1974), Chemical applications of lasers reviewed by Jones
(1969) and Haught (1968). See Pressley (1971) for a com
pilation of data on lasers, and Levine and OeMaria (1 966 et
seq.) for recent advances.
level. In atomic structure and spectroscopy the name level
denotes a particular value of the total 'angular momentum J.
For example, from the "configuration 1s 2 2s 2 2p arises a
doublet P term written J P. The two levels of this doublet
3 1
correspond to J = ^ and J = ^> * or only these a n 9ular
momenta may be constructed from 1 = 1 and S = \ (see
"angular momentum). The two levels of the term therefore are
written 2 P, /2 and ^P^.
The number of levels of a particular term is its
"multiplicity. Thus in 2 P there are two levels, and in 3 there
are three levels ( 3 D t , 3 D 2 , and ?D 3 ). The levels differ in energy
because of the °spinorbit coupling, and their order can be
predicted on the basis of the Hund rules. The angular
momentum./ may have 2/ + 1 different values of its
component on some arbitrary axis; in common with the
practice of angular momenta these are distinguished by the
value of Mj. The state of a level of a term is denoted 2 ? y J ,
etc.
Questions. Define the use of the term 'level' in atomic
spectroscopy. Write down the levels that may arise from the
following terms: l P, 3 f, 3 S, 4 D, 6 D. Under what circumstances
does the superscript not denote the mutiplicity? (See "term
symbols.)
Further information. See Chapter 8 of MQM for more detail
about atomic spectra. Levels, multiplicity, and fine structure
are discussed by Kuhn (1969) and Herzberg (1940). Turn to
•atomic spectra and fine structure for further information.
ligandfield theory: a synopsis. The ligandfield theory of
the structure of complexes of transitionmetal ions is an
s.Q) %o>
FIG. L4. The 6 ligand orbitals and the 6 sy m me iry adapted
COmbi nations.
128
ligandfiekl theory: a synopsis
extension of the 'crystal field theory which takes into
account the known derealization of the electrons of the ion
into the orbitals of the surrounding ligands; at the same time it
makes use, like the crystalfield theory, of the very high degree
of symmetry of the complexes normally encountered. The
ligandfield theory is essentially a 'molecularorbital theory of
complexes, and begins by concentrating its attention on the
delectrons of the central ion. We shall illustrate the method by
considering an octahedral complex in which the central ion
possesses n delectrons.
Consider the ligands as bearers of (Jorbitals which approach
the central ion with a lone pair of electrons. We denote these
by the spheres in Fig. L4. From these six orbitals six combin
ations may be formed: the six chosen have welldefined
symmetry properties {For those who know "group theory, we
select "linear combinations that span irreducible represen
tations of the octahedral point group.) The six combinations
are illustrated in the figure: they fall into three groups. Only
one of the groups, that labelled e, has a net "overlap with the
dorbitals of the central ion, and so only this combination can
form bonding and "anti bonding molecular orbitals with the
dorbitals. It follows that a molecularorbital diagram of the
type shown in Fig. L5 may be anticipated. The energies of the
ligand and ion orbitals are such that the lowerenergy combin
ation is largely ligand in nature, and the antibonding combin
ation is nearly metal ion in character.
liqandi
odd n elections
complex
largely metal
character
i largely ligand
character
FIG. L5. The figandfield splitting in an octahedral complsK.
Into this set of eleven orbitals (a doubly "degenerate
bondingorbital labelled e , four degenerate nonbonding
orbitals confined entirely to the ligands and labelled t iu and
a lg , three triply degenerate nonbondingorbitafs confined
entirely to the metal ion and labelled t^, and an antibonding
combination labelled e*) we must insert [12 + r?) electrons
g
(2 from each ligand and n from the ion). Applying the 'aufbau
principle it should be clear that the first 12 electrons will
occupv the orbitals e , t , a , and that these are of predomi
r * g lu' jg'
nantly ligand character. The next/i electrons have to compete
for places in the orbitals t„ and e*: this is just the situation
2g g
encountered in the "crystalfield theory and Figs. C6 and C7
should be consulted; there we see that if the energy gap
between the orbitals is large {the strongfield case) all the
electrons attempt to enter the t set, and enter the upper e
set only if the "Pauli principle forbids them entry into the
lower set. If the orbital separation {there called lODq) were
small (the weakfield case) it might be energetically favourable
for the electrons to enter the t and the e* orbitals, but to do
2g g
SO with their spins parallel ( "Hund rules).
The distinction between the strong and weakfield cases,
and their generation of low and highspin complexes, is
carried over from the crystalfield theory into the molecular
orbital theory: the difference lies in the source of the splitting.
Another difference is the fact that the e*orbital is not wholly
g
confined to the metal ion: since it is formed from the overlap
of metal and ligand orbitals it contains some ligand character.
This means that any electrons that occupy it may spread over
on to the ligands. The evidence that this happens comes from
spectroscopy, especially "electron spin resonance (where
"hyperfine structure due to the ligands has been observed).
The other important improvement of ligandfield theory
over crystalfield theory is the natural way that the former
permits itbonding between the metal and the ligands. This
is especially important when the ligands are species such as
CO or NO. In order to see the effect of permitting Trbonding
consider the octahedral complex again, and this time, in
addition to the abonds, let each ligand possess two orbitals
that are perpendicular to the rnetalligand bonds. From these
12 orbitals, which are illustrated in Fig. L6, 12 combinations
may be constructed. Three of the combinations have the same
symmetry as the d
xy
d ,d orbitals of the central ion
xz yz
linear combination of atomic orbital 5 ( l.CAO)
129
FIG. L6, The 1 2 jrorbrtals of octahedrally disposal) Ngands.
FIG. L7, irbonding: the effect on energies. In (a) the ligand 7rlevels
are initially full; in (.b) they are empty.
{which so far have formed the nonbonding t set of orbitals
confined to the metal). When we permit this overlap to occur
the molecularorbital energylevel diagram is modified. Two
situations need to be distinguished: in the first the TTorbitals
of the ligands are full (and lie below the tforbitals); in the
second they are empty (and iie above the rjorbitals). The two
cases are illustrated separately in Fig, L7. In the former case
the ligands bring up their 7Te!ectrons and fill all the bonding
combinations of Trorbita!s. This leaves the n electrons from the
metal to be distributed between the antibonding it combin
ation (t* ) and the untouched e* combination. Since the
previously nonbonding orbitals nave become slightly antibond
ing, the gap (10Dc/) has been reduced by the presence of the
TTelectrons. The opposite is the case when the 7Torbitals of the
ligands are empty, for now the n electrons compete for places
between the bondinq t combination and the antibonding e*
29 9
combination. Thus the splitting has been increased by the
presence of unfilled irorbitals.
The size of the splitting lODq determines the spectroscopic,
magnetic, and chemical properties of the complex in the same
way as there are determined by the "crystal fie Id theory, but
the distribution of the electrons on to the ligands gives them
some of the latter's "spinorbit coupling energy.
Further information. See MQM Chapter 9 for an account of
ligandf ield theory. A simple introduction to the Ideas of
ligandfield theory is provided by Earnshaw and Harrington in
The chemistry of the transition elements (OCS 13); this is
developed further in Coulson's The shape and structure of
molecules (OCS 9), The effect of these ideas on the explan
ation of the magnetic properties of complexes is described by
Earnshaw (1968). For spectral consequences see Jorgensen
(1962, 1971). A simple and good introduction to ligandfield
theory has been given by Orge! (1960), Coulson (1961), and
Murrell, Kettle, and Tedder (1965), and developed more
mathematically by Figgis (1966), Ballhausen (1962), and
Griffith (1964). See also crystal field theory and the °Jahn
Teller effect.
linear combination of atomic orbitals (LCAOh An
LCAO is a method of describing a molecular orbital covering
several nuclei in terms of a sum of atomic orbitals centred
130
linear combination of atomic orbitals (LCAO)
on each nucleus. Thus the bonding orbital in the hydrogen
molecule is really a complicated function spreading round
both nuclei, but since it is expected to resemble the 1satomic
orbitals on each nucleus the orbital is expressed as the LCAO
formed by the "superposition of the two Isorbitals (Fig. L8).
that lies deeper than the mathematical device of being able to
expand any function in terms of a sufficiently complete set of
simpler functions. This deeper justification is provided by the
"superposition principle. When one has the possibility that a
variety of processes can occur the superposition principle
IG. L8. In the LCAO method the wiavefunctions of the atoms (black)
are superposed to give a molecular orbital (colour).
If all the atomic orbitals were used in order to reconstruct
the true molecular orbital the LCAO procedure would be
exact, but this task is too heavy and normally a small number
of atomic orbitals is selected as the basis set; used in this way
the LCAO procedure is an approximation. In the elementary
discussion of the "hydrogen molecule the basis set consists of
the two 1satomic orbitals, and this gives a reasonably good
description of the bond, but it can be improved considerably
by expanding the basis to include the 2s and 2porbitals, and
others. The use of a small basis set in the LCAO description of
molecular bonding is one of the gravest sources of error in the
method, and the selection of too small a set can make a
nonsense of an otherwise elaborate calculation.
The use of the LCAO method has a theoretical justification
demands that one should consider the probability amplitudes
for the individual processes rather than the probabilities
themselves. The total, composite process is described by the
total, composite amplitude, and all manipulations and thought
must be applied to this object, and the probability for an
individual process extracted only at the end of the calculation.
In the case of a molecule we have an example of such a
composite situation, for there is a probability that an electron
is on one of a number of nuclei. Our problem is the distribution
of this electron in the face of these various possibilities, and
the superposition principle tells us that in order to find this
distribution we should construct the overall amplitude
(molecularorbital wavefunction) by superimposing the
amplitudes of the individual processes (the atomicorbital
localized orbitals
131
wavef unctions). But this is precisely the line taken by the
LCAO method in its attempt to construct the molecular
orbital from the individual atomic orbitals.
Questions. What is the LCAO procedure? Is it accurate? What
is the worst source of error in the method? Can the method
be justified? What is the basis set in the simplest description
of molecular oxygen (that is, we require the minimal basis set)?
How cart a simple description of molecular hydrogen be
improved? What guidance can the physical nature of the
problem provide in the problem of extending the minimal
basis set? (Think about energy, size, and orientation of the
orbitals, and how the polarization of an atom by neighbouring
nuclei can be taken into account.)
Further information. The LCAO method is at the foundations
of molecularbonding theory, and more information will be
found under appropriate entries. See in particular Coulson's
The shape and structure of molecules (OCS 9) and MQM
Chapter 9. See also Coulson (1961), Murrell, Kettle, and
Tedder (1965). and McGlynn, Vanquickenborne, Kinoshita,
and Carroll (1972). For a simple introduction to °self
consistent field calculations on molecules see Richards and
Horsley (1970). The choice of orbitals to combine into a
particular molecular orbital must conform to the symmetry of
the molecule, and "group theory can be extremely useful for
determining appropriate combinations. See MQM Chapters 5
and 9, Cotton (1963), Tinkham (1964), and Bishop (1973).
localized orbitals. The chemistry of many molecules suggests
that to a significant extent electrons may be regarded as
belonging to different parts of the molecule; "molecular
orbital theory gives the impression of predicting that all
electrons are spread throughout each molecule, and therefore
it seems to run counter to the chemical evidence. The
deficiency is apparent rather than real. It is possible to
manipulate the form of molecular orbitals (by taking various
sums of them) and to generate localized orbitals, orbitals that
are localized almost wholly in the vicinity of different groups
of the molecule. This procedure is illustrated in Fig. L9 (which
is based on the calculation in Question 2),
It is possible to take the appropriate linear combinations
because the actual manyelectron wavefunction of a molecule
basis ,
orbitals
locofe«i orbitals
e
V
w
&
M&.
fully localized orbitak
>u=A
FIG. L9. The formation of localized orbitals. Only when m
localization obtained.
X is full
must satisfy the °Pauli principle, and this is ensured if it is
written as a 'Slater determinant. It is an elementary feature of a
determinant that rows or columns may be added to other
rows or columns without changing its value. Therefore similar
manipulations applied to the Slater determinant do not change
the total wavefunction, and the localized description is
mathematically equivalent to the original delocalized descrip
tion. An example of the procedure is set as a Question.
Questions. 1. What is the aspect of the molecularorbital theory
that seems to conflict with chemical evidence? What evidence
supports the view that electrons are indeed delocalized
throughout the molecule? How are localized orbitals generated?
What do they succeed in achieving? Why is it a permissible
procedure? Why are delocalized and localized descriptions
equivalent? Suppose we have an orbital which is a sum of an
sorbital on atom a, an sorbital on atom b, and an sorbital on
atom c, and we call this \ii u then the orbital is tt. = c s +
r ' a a
c b s b C eV" and let there ^ a similar orbital in which the
132
lone pair
central atom b contributes a porbital— then the bonding
orbital is \Jj 2 = c s + c'a, + c s . Show that the combin
ations 0] + 2 and i^i — \p 2 are largely localized in the
a— b and b— c regions respectively, and sketch the resulting
localized orbitals. The reason why it is correct to take these
combinations is treated in the next, harder Question,
2. Consider a linear triatomic ABj molecule (BAB), and let the
occupied orbitals be of the form ipt — s + M(a + b) and
02 = P + ^fa — b) (for the notation see Fig. L9), These
orbitals contain four electrons, and so the antisymmetrized
wavefunction is the "Slater determinant ip= (1/4!)^
l^ iQ (1)0 (2(i// {3)0 (4) I, where a and denote the spins.
Show that this determinant may be manipulated, without
change of value, into the determinant ip = (1/4! ) Yl
li/£(1) 0*(2) a (3)t//j(4) I, where 0*= (1)^(01 ± 2 ). Form
these orbitals from the original pair and show that they are
localized, but that the localization is complete only in the case
where u = X. The procedure is illustrated in Fig. L9. Discuss
your result in terms of the "hybridization of the central atom.
Further information. See MQM Appendix 9.2 for an account of
localization and a worked example. See also Coulson's The
shape and structure of molecules (DCS 9) and Coulson (1961).
See also Murrel!, Kettle, and Tedder (1965), Slreitweiser
(1961), Salem (1966), and Pilar (1968). Good accounts in the
literature on the formation of localized bonds are those of
LennardJones and Pople (1950), Boys (1960), and Edmiston
and Ruedenberg (1963, 1965).
lone pair. A lone pair of electrons is a pair of electrons of the
valence shell not engaged in bonding. As an example, consider
the tetrahedral distribution of electrons around the oxygen
atom in HjO: two electrons stick one proton to the atom, two
stick the other, and the remaining four form two lone pairs
sticking out like rabbit's ears on either side of the molecular
plane. We may consider the last four to be electrons in
localized, nonbonding orbitals.
Lone pairs are important both structurally and chemically.
They influence the structure of a molecule by exerting large
repulsive effects on the electrons in neighbouring bonds: for
example, the pyramidal shape of ammonia may be traced
partly to the effect of the single lone pair on nitrogen
exerting a repulsive force on the six electrons in the three
N— H bonds. This analysis of molecular structure in terms of
the lonepair interactions is the basis of the SidgwickPowell
rules which state that tone pairs dominate the repulsive inter
actions in molecules. In chemistry the lone pair is a nucleo
philic centre because it can so readily form a dative bond to an
electropositive centre: lone pairs act as a base (in the Lewis
sense).
Since lone pairs are not tied into place by a parasitic
nucleus they also contribute strongly to spectra, and the
it** — n (read 'n to pi star') transition in carbonyl compounds
is a major mode of excitation (see colour). The n stands lor a
lonepair (nonbonding) orbital, and the transition takes the
electron from the oxygen lone pair and spreads it over the
carbonyl group by depositing it in the antibonding JTorbital.
Note that this transition changes the charge distribution in the
carbonyl group and so is strongly responsive to solvent effects.
Further information . See MQM Chapter 10. For a discussion of
the SidgwickPowell rules and their modern development see
Bader (1972). For references to the role of lone pairs in
spectra see "colour.
M
magnetic dipole and electric quadrupole radiation.
The most intense transitions in molecules are due to electric
dipole transitions, but when these cannot operate (when they
are forbidden by 'selection rules) other mechanisms may have
sufficient strength to cause a transition, albeit at a much lower
intensity. One such mechanism is the magnetic dipole tran
sition, which relies for its operation on the magnetic com
ponent in the light field; the other mechanism, of similar
strength, is the electric 'quadrupole transition, which relies
for its operation on the variation of the electric field of the
light over the space occupied by the molecule (that is, there
must be a field gradient on a molecular scale if the quadrupole
transition is to operate).
The magnetic dipole transition generates the magnetic
component of a light field just as an electric dipole transition
generates an electric component {but, of course, in each case
the other component is forced to accompany the generated
component). There are two principal differences.
The first is the weakness of the interaction of the molecule
and field via the magnetic dipole. Pictorially, this can be traced
to the rotational nature of the magnetic dipole transition: if
during a transition charge is displaced in a curved path it wilt
possess a magnetic transition dipole (Fig. M1). But in a region
as small as the extent of a molecule the curvature of the dis
placement will be only weakly apparent: if D is the diameter
of the molecule and Xthe wavelength of the emitted or
absorbed light, it is plausibie to suppose that the efficiency of
the coupling is of the order of DfK Since the intensity of a
transition is proportional to the square of its "transition dipole
moment, this suggests that the intensity should be only about
{D/X) 2 of the intensity of an electric dipole transition, where
curvature need not be detected. For typical molecules and
wavelengths (D/X) 2 ~ 10~ s , and this is the order of magnitude
of the intensities observed in practice.
The other difference between electric and magnetic dipole
transitions is in the selection rules: a magnetic dipole transition
is akin to a rotational displacement of charge; a rotation is not
reversed when it is inverted through a point (Fig. Ml); there
fore a magnetic dipole transition has even parity (see °gerade
and ungerade\. Unlike an electric dipole transition (which is a
translation of charge, and which is therefore of odd parity,
electric dipole
transition
magnetic dipole
transition
\ I I J > J electric quadrupol
transition
FIG. M1 . The charge displacement in transitions of different type.
133
134
magnetic moment
and where the selection rules AL = ± 1 and g — ► u emerge) in
a magnetic dipole transition AL = 0, g — * g, and u — ► u. The
overall selection rule AJ= ± 1 applies to both the electric and
magnetic dipole transitions, because both are dipolar. (See
'■electric dipole transition, and the account of the role of
photon angular momentum.)
The electric quadrupole transition arises from a displace
ment of charge that has a "quadrupolar nature. This somewhat
subtle (but simple) type of charge displacement, which is illus
trated in Fig. Ml, can be detected with an efficiency of the
order of D 2 1 A 2 (in intensity), and so we expect magnetic dipole
and electric quadrupole transitions to be of comparable inten
sity. But detailed analysis shows that the latter also depends on
the square of the frequency, and for visible light the intensity
is reduced by a further two orders of magnitude. Therefore, an
electric quadrupofe transition has an intensity of only 10" 7 that
of an electric dipole transition.
Since we are now dealing no longer with a dipole but with a
quadrupole, the selection rules differ. A quadrupole can be
envisaged as two dipoles in opposition: therefore we expect it
to be of even parity. Consequently g — * g, u —* u transitions
are allowed. More detailed analysis shows that the angular
momentum selection rules are AL = 0, ±1 , ±2 and AJ = 0, ±1 , ±2.
One might ask what has happened to our arguments concern
ing the unit spin of the 'photon? The answer lies in the quad
rupolar nature of the transition: as the photon is flung off the
radiating molecule the spatial variation of the transition
endows it with an orbital angular momentum. The total angular
momentum of the photon may exceed unity, and so a selec
tion rule of AJ = ±2 can still be understood in terms of the
conservation of angular momentum.
Further information. Magnetic dipole transitions are nicely
discussed in §IIID 2b of Kuhn (1962), in §3.2.2 of Griffith
(1964), and in §7.6 of Hameka (1965). See also Heitler (1954)
and Berestetskii, Lifshiu, and Pitaevskii (1971). The tran
sitions that are responsible for "electron spin resonance and
"nuclear magnetic resonance, where a magnetic moment couples
with an oscillating electromagnetic field, are important
examples of magnetic dipole transitions. If a molecule can be
excited to the same state by an electric and a magnetic dipole
transition it is optically active (see "birefringence),
magnetic moment. The magnetic moment of an electron due
to its "orbital angular momentum (which may be pictured as
arising from a circulating current) is u, = 7 I, where 1 is the
orbital angular momentum and 7 n a constant of proportion
ality known either as the magnetogyric ratio or as the
gyromagnetic ratio (the former name is more helpful).
Simple calculation shows y s to be equal to — e/2m ;
the negative sign of 7 (which arises from the negative charge
of the electron) shows that the direction of u. is opposite to
the direction of 1, but they are collinear (see Fig. M2).
piofon
, MC 9^
Af^/V**
FIG. M2. Orbital and spin magnetic moments.
The "spin angular momentum also gives rise to a magnetic
moment (as a simple picture of a rotating charge would suggest)
but its magnitude is 'anomalous' (which means that people
could not explain it when it was first encountered), and in
order to express the moment in terms of the magnetogyric
ratio 7 an extra factor (the "g factor, is introduced: then
jJ^ = gy s. Experiment, and later theory, showed thatff =
2 , 0023, and this is frequently approximated as 5 = 2. The spin
magnetic moment and the spin momentum are collinear but
anti parallel.
The magnetic moment is often expressed in terms of the
Bohr magneton U = eh 12m , which is a positive quantity and
may be considered to be a basic unit of magnetic moment (its
magnitude is 9273X 10
J T 1 , or 9273 X 10T 2 * JG"'). In
U fh and so
terms of the magneton we have 7
M L = (P B /h)l and n s  "s(/u B /li)„.
Just as the electron is the elementary negative charge and
representative of the lighter fundamental particles (the
magnetic properties
135
leptons, but note that the class lepton contains neutrinos and
muons) and £l B is the elementary unit of magnetic moment, it is
convenient to consider the proton as the elementary positive
charge and representative of the heavier fundamental particles
(the baryons, the proton and neutron, a subset of this class,
are called nucleons). The corresponding elementary unit of
magnetic moment is the nuclear magneton , /u N = eh/2m p , and
it has the magnitude 5051 X ICT^JT'or 5051 X 10" 3I JG 1 .
The enormous difference in magnitude between the Bohr and
nuclear magnetons is due to the difference in mass of the
electron and the proton (to achieve the same spin angular
momentum as an electron, the heavier proton needs a much
smaller angular velocity, and so the equivalent current loop
carries much less current, and the magnetic moment is smaller).
The nuclear magneton is of a size convenient for the expression
of the magnetic moments of nuclei (and, incidentally, of the
magnetic moments of rotating molecules: see ^value). If a
nucleus has a spin angular momentum I its magnetic moment
isff n (M N /h)l, where the nuclear gvalue, which depends on the
nucleus, is defined by this relation and is determined by
experiment (and in the future, one hopes, by calculation in
terms of the nuclear structure). It is found that g may be
n
positive or negative, depending on the element and the isotope,
and it should be food for thought to be told that even the
uncharged neutron has a magnetic moment. Typical values of
the magnetic moments of some common nuclei are listed in
Table 17. Notethat? =g (Ji„,/h).
Further information. See MQM Chapters 8, 9, and 10 for a
discussion of magnetic moments. The classical electromagnetic
theory concerning magnetic moments is well described in
Corson and Lorrain (1970). The Maxwell equations are sum
marized in Table 20. The quantummechanical theory of
magnetic moments is described further under "spin, Rvalue,
and "magnetic properties. See also "Dirac for his contri
bution. See 'electron spin resonance and 'nuclear magnetic
resonance for one way of harnessing magnetic moments into
useful employment.
magnetic properties. When a substance is immersed in a
magnetic field it has an effect which may be visualized
(Fig. M3) in terms of the distortion of the lines of force of the
FIG. M3. A schematic indication of the magnetic induction (magnetic
flux density) in magnetic materials of different kinds.
field. In a diamagnetic material the number of lines of force is
reduced (or the magnetic induction B is less within the body
than in free space), and in a paramagnetic material the number
(and induction) is increased. An alternative way of expressing
this behaviour is to regard a diamagnetic sample as magnetized
in opposition to the direction of the applied field, and so to
give rise to an opposing field which partially cancels the applied
field; conversely, a paramagnetic sample is magnetized in the
same direction as the applied field, and the field that this
induced moment generates augments the applied field. A
physical manifestation of paramagnetism and diamagnetism is
that a paramagnetic sample tends to move into a magnetic field,
and a diamagnetic sample tends to move out of it.
The ratio of the induced magnetic moment to the strength
of the applied field is the magnetic susceptibility of the sample.
136
magnetic properties
BOX 12: Magnetic properties
Magnetization M = X H
X is the magnetic susceptibility and H the field strength,
y m vP + Y d
A m A m A m
paramagnetic susceptibility X p >
diamagnetic susceptibility X^,< 0.
Magnetic induction (flux density)
B = HoH + fiolVl = Mo(1+xJH.
Curve /aw
X m = C/F; C = (ioL(i 2 m
(u. is the magnetic moment; for example, (i 2 ~ ^V^StS + 1)
for spinonly paramagnetism).
CurieWeiss law
X m = CHT~0).
Brillouin function
M = NnSjiliB/kT)
Perturbation theory expression for the molar
susceptibilities:
TIP: x p
ZeVjA y , j (0llln)(nim» j
Xj), = i l^ 2 ) <*" 2 > (LartgevinPauli
equation)
See Table 20 for the Maxwell equations.
and it is normally denoted /^ (see Box 12 and Table 20). For
a diamagnetic material the susceptibility is negative, and for a
paramagnetic material it is positive (in accord with the differ
ent directions of the induced magnetization). The susceptibility
of all materials can be written as the sum of a paramagnetic
susceptibility x^ and a diamagnetic susceptibility x^:
X m = X^, + X^ (remember that x^ < and that for most
molecules the diamagnetic term dominates the paramagnetic).
When the molecule possesses unpaired electrons the paramag
netic susceptibility dominates, and it is generally found that }P
diminishes as the temperature is raised and that in the vicinity
of room temperature it is proportional to 1/7" (the Curie /aw).
In a few cases the paramagnetic term dominates the diamagnetic
even though all the electrons are paired; in such cases it is also
found that this weak paramagnetism is independent of tempera
ture; for this reason it is referred to as temperatureindependent
paramagnetism {TIP} (an alternative name is highfrequency
paramagnetism) ,
1 . Dtamagnetism . Al! molecules have a diamagnetic
component of their magnetic susceptibility which arises by the
applied field exerting a torque on the electrons present. The
torque tends to drive the electrons in circles within the molecu
lar orbitals (Fig. M4 a), and the circulating current so produced
FIG. Ml, (al Dtii magnetic and (b) paramagnetic currents. In la) The
field drives a current uvithin atomic orbitals; in (b) it excites it through
the molecular framework by mixing in excited states.
sets up a magnetic moment and field in opposition to that
applied. Because the diamagnetic susceptibility arises from
processes happening within the ground state of the molecule,
Pascal was able to draw up a table of contributions to the total
diamagnetic susceptibility of a molecule in terms of its" struc
tural features which could be transferred between molecules.
Aromatic molecules, in which there is a cyclic path for the
electrons of the Ttsystem, show anomalously large suscepti
magnetic properties
137
bilities, which are ascribed to ring currents. These ring currents
are of particular importance in determining the form of
"nuclear magnetic resonance spectra, but there has been some
quarrel over their existence.
2. Paramagnetism, The spin paramagnetism is easily under
stood in terms of the "magnetic moment associated with the
electron's "spin angular momentum. In a magnetic field the
energy of a magnetic moment depends on its orientation;
since an electron may have one of only two orientations with
respect to a selected axis, the application of a magnetic field
to a collection of molecules, each with a single unpaired spin,
lowers the energy of those with j3spin (m = ~) and raises
by an equal amount those with aspin {m = + ^ Ver V
quickly the collection of molecules relaxes into thermal
equilibrium, and the sample then contains more £spins than
aspins, (Fig, M5)(the proportions are determined by the
ttfttfftff poromoqnetc, no field
f 4 tff4$T$4 paramagnetic, field present
tHHtttit ,M *™<*«
ifftfftftt K ' eal ontiferromoqnet
FIG. MS. Magnetic materials dependent on electron spin.
Boltzmann distribution). The spin magnetic moment lies in a
direction opposite to the spin angular momentum (because
the electron Is negatively charged), and so at equilibrium there
is a net moment parallel to the applied field. The field it
generates augments the applied field and the sample is
paramagnetic. As the temperature is raised the Boltzmann
distribution becomes more even between the two spin orien
tations because the thermal motion jostles the spin alignment,
and consequently the net induced moment falls; a simple
calculation applicable to room temperatures (see Question 2)
leads to the Curie law, and a simple extension that leads to an
expression valid at all temperatures may also be deduced {so
long as the material does not turn ferromagnetic, see below):
this is the Briltouin function. Both these expressions may be
calculated for molecules with arbitrary spin. [For instance,
molecular oxygen is a paramagnetic species with S = \, and so
there are three orientations with different energies.)
An important point about the role of the magnetic effects
of the orbital angular momentum should always be remem
bered: in many molecules this motion is °quenched and so
makes no contribution to the magnetic susceptibility— the
magnetic behaviour is then referred to as spinonly
paramagnetism. When the orbital motion is not fully
quenched the situation is more complicated; so too is it when
the "spinorbtt coupling energy is large, and then the magnetic
susceptibility cannot be calculated simply by counting spins
and applying the Brillouin or the Curie formulas.
Temperatureindependent paramagnetism is a property of
the "orbital angular momentum of the electrons. If there are
low lying excited states the magnetic field can make use of
them to induce a migration of electrons through the molecular
framework, and the orbital angular momentum of this motion
gives rise to an orbital magnetic moment. This orbital motion
differs from the diamagnetic current in so far as it arises from
a distortion of the electronic distribution by the field, whereas
the diamagnetic current occurs within the undistorted
molecular orbitals. This difference leads to a current which
percolates through the molecule in the direction opposite to
the diamagnetic drifts within the molecular orbitals (Fig. M4),
and so it gives rise to a magnetic moment that enhances the
applied field. The magnetic field determines the sense of
circulation of the current, and therefore the direction of the
magnetization. It follows that the latter is independent of the
jostling motion that randomizes the spin magnetic moments,
and so this paramagnetism is temperatureindependent.
3 . Ferrom agne tism and an tiferromagne tism .Atsufficently
low temperatures many paramagnetic materials undergo a
transition to a state where all the spins align cooperatively
and strongly enhance the magnetic properties of the material
(Fig. M5). When the neighbouring spins align in the same
direction throughout a reasonably extensive region (a domain)
138
magnetic properties
of the sample, a strong magnetization is obtained and remains
after the magnetizing field is removed. This is the ferromagnetic
phase, and a common examples are those of iron with a transition
temperature {known as the Curie temperature) of 1043 K, and
of cobalt, with a Curie temperature of 1403 K (these high
temperatures explain why 'permanent' magnets are made of
these materials). Their magnetic susceptibility above the
Curie temperature follows the CurieWeiss law (Box 12), in
which the 1/7" of the Curie law is replaced by 1/(7" +T), 7" c
being the Curie temperature (more accurately, 1/(0^+7"), where
8 Lis the paramagnetic intercept, which is slightly larger than
T c : for Fe, Q Q = 1100 K and for Co, 8 Q = 1415 K). Another
group of materials, of which NiO is a famous but far from
unique example, shows a transition to an antiferromagnetic
phase, in which neighbouring spins are aligned in opposition
(Fig. M5) ; therefore the spin paramagnetism is strongly
quenched in this cooperative state. The transition tempera
ture is known as the Neet temperature. The spinspin
interactions responsible for the alignment in both kinds of
phase are electrostatic in origin and related to the "exchange
energy; see Further information.
Questions. 1. How could you recognize diamagnetism and
paramagnetisim in a material? low is the susceptibility related to
the induced moment? What are the dimensions of Y ?
Consider the work necessary to insert a magnetic sample into
a magnetic field; which is easier to insert, a paramagnetic or a
diamagnetic sample? What is the source of paramagnetism?
What is the difference in energy between 1 mol of electron
spins in the ^orientation and 1 mol in the aorientation when
a field of 5 kG (0'5 T) is present? What is the population
difference when the spins are in thermal equilibrium at 300 K?
What is the magnetization of this sample, and what is its
susceptibility? [Try to get your units right: magnetization has
units of A m , so does the field strength; magnetic induction,
or flux density, has units of Wb rh" 2 , or kg s" 2 A 1 , or T; see
Table 20.) What is the difference between the currents giving
rise to diamagnetism and those giving rise to TIP? How can you
justify Pascal's rules? What is the difference between
ferromagnetism and antiferromagnetism? What are the tran
sition temperatures called in each case? What is the nature of
the interaction between the spins that gives rise to the co
operative phenomena?
2. Deduce the Curie law as follows. The magnetization
(magnetic moment) of a sample in a magnetic field B is equal
to the magnetic moment of an aspin multiplied by the
number of aspins, plus the moment of a fSspin multiplied
by the number of 0spins. The numbers of <xand j3 spins are
determined by the Boltzmann distribution at a temperature 7,
the energy of the spin with magnetic moment m being ~m^S
as the field lies along ^. When this energy is small the
exponential in the Boltzmann distribution may be expanded.
Your answer should be the Curie law exhibited in Box 12.
3. Repeat the calculation for a genera! spin S, and find the
expression for the temperature dependence of the magnetiz
ation at all temperatures: this is the Brtllouin function. The
answer is shown in the Box. Plot this function's dependence
ong}i a BlkT.
o
Further information .The standard work on magnetic
susceptibilities is by van Vleck (1932), but as it was written
so long ago it uses rather oldfashioned language. See MQM
Chapter 1 1 for an account of the calculation of magnetic
properties and their interpretation in terms of currents.
Earnshaw's book (1968) gives a discussion of many of the
points mentioned in the preceeding paragraphs, and other
volumes of interest are those by Oavies (1967) and Selwood
(1956). A good introduction to cooperative phenomena is
provided by Stanley (1971), and a tough but good intro
duction to cooperative magnetic phenomena is given by
Mattis (1965). Much interest in magnetic properties arises
from the application of "ligandfield theory to transition
metal ions; therefore you should look at Earnshaw and
Harrington's The chemistry of the transition elements
IOCS 13), Earnshaw (1968), Orgel (1960), and
Griffith (1964). More information about magnetic properties
will be found under "electron spin resonance, "nuclear
magnetic resonance, and "chemical shifts. The controversy
over the existence of ring currents may be traced by referring
to Musher (1966). An excellent compilation of magnetic
properties of a wide variety of materials is in §5 of Gray
(1972), The Maxwell equations are summarized in Table 20.
matrix
139
matrix. A matrix is a rectangular array of numbers. It may be
regarded as a generalization of the concept of 'number' in the
sense that an ordinary number is a 1 X 1 matrix and is therefore
a special case of the general n X m matrix of numbers. In an
n x m matrix (which we denote M} the numbers that constitute
it (the matrix elements) may be labelled according to the row
and column they occupy. Thus each element may be denoted
M fC : the first subscript labels the row, the second the column.
In a square n x n matrix the number of rows is equal to the
number of columns, and there are n 2 elements. Square matrices
are very important in quantum mechanics (although oblong
matrices also occur), and our comments will be confined to
them.
BOX 13: Matrices
A square matrix M is the array of n 2 elements M fc : M K is
the element in row r, column c:
M u "*12 ^13 ... /Wi„
/W 21 Af 22 M23 ■ ■  M 2 „
Ms
L M nl M r>2 M t»3 ■ ■ • M nnJ
Addition M + N = P, where P = M„ + /V,„.
' IC Fc fC
Multiplication MM = P, where P fe = ^ M N .
Examples: if M
"♦"ft;.*;;]'* (::££:)
n.b. NM=[" + * ?* + C 'l # MN in general,
[ah *dg bh + cg_
Special matrices (and illustrations using the example of M
defined above}
L
Diagonal matrix
A„  unless r = c, for example, A =
a
Unit matrix
1
all elements on diagonal = 1
all elements off diagonal = '
_o ij «
often denoted 8 and
re
for example, 1
_U ij
then called the Kronecker delta
Inverse matrix M 1 : MM 1 = NT'M = 1 (see below)
Transposed matrix
M: M rc = M cr , for example, M =
Complex conjugate
M*: [M') rc = W re )\ for example, M*  fjl *.*! .
Adjoint matrix Ivf: M* = M*; that is Mf*), c = M* cr .
for example, M* = ,. . I .
c J
Unitary matrix M* = M" 1 ,
Hermitian or self adjoint matrix M* ■ M.
Determinant of matrix: iMl or det M;
for example, fM I = ac — bd;
n.b. if P = MN, P\= IMl INI.
To find the inverse.
1. Find IMl; if lMl# Othe inverse may be found; if IMl
= the matrix is singular and has no inverse.
2. Form M .
3. Form M', where the element M' is the cofactor
— re
(the signed minor) of W) fC 
4. Form the matrix M'/lMl; this is the inverse M 1 .
For example, 1 . IM I = ac — bd,
3.
■'"■til]
^[=4] [4 1]
(contd.)
140
matrix mechanics
= / 1 \ facbd ab + abl
\acbdj [cdcd bd + ac]
= 1 = M" 1 M .
To solve a set of linear simultaneous equations
If the equations for n unknowns are
M u x , + M t2 x 2 + ... M in x n = cy
M 2i x 1 + Af M x 2 + , . . Af 2 „x„ = c 2
Af,,^! + M„ 2 Jf 2 + . . . M nn x n =c n
write them as Mx = c where x and c are the n X 1 matrices
x =
Xi
■*«r
x 2
c =
Cj
>_
c «
Then as M~'M = 1 and 1x = x
x = M _1 c.
Therefore, find M" 1 by the preceding rule, and form M"'c
to find then unknownsx 1( ,. .x ,
Matrices may be combined together by following certain
simple rules. If two matrices have the same dimension (number
of rows or columns) they may be added and multiplied
together: the rules for each type of combination are given in
Box 13. Some matrices have special properties: these are also
given in Box 13. Note especially that the rule of matrix
multiplication differs from that for the multiplication of
ordinary numbers {or cnumbers as they still are occasionally
called): for matrices it is not generally true that the product
MN is equal to the product NM: matrix multiplication is non
commutative in general. The difference MN— NM is known as
the commutator of M and N: the fact that the commutator
does not necessarily disappear leads to the most significant
differences between matrices and ordinary numbers, and, at a
different level, is a manifestation of the differences between
classical and quantum mechanics: see "matrix mechanics.
1 2
3 4
B =
5 6
7 8
C =
3 2 + 41
5 6
The language of matrices has spread into ordinary
Schrddinger quantum mechanics, for there one encounters
integrals over wavef unctions having the form J"dT^*li^ c ,
where SI is some "operator. A convenient notation for this
integral is £l rc , and ail the objects that may be formed from
the wavefunctions of the system, where the labels r and c run
over all its states, may be arranged into the matrix fi (see,
for example, 'perturbation theory).
Questions. Many of the techniques of matrix algebra may be
illustrated with twodimensional square matrices. The
following simple problems are based on the three matrices
A
1. Identify the elements A u , A u, fl 21l C n , C n .
2. Form A+B, A B, A I B+C, AB, BA, A(BC), (AB)C, ABBA,
ACCA,
3. Form A, A + , A*, A" 1 , det A, C\ C + .
4. FormB" 1 AB,B~ 1 B, BB~ 1 .
5. Using the rule for matrix multipHcation and identifying the
2 X.I matrix x as[*l , express the set of simultaneous linear
equations x \ 2y 2 and 3x f 4 y = 5 as a matrix equation.
This equation will be of the form Mx=N; show that x and y
may be found if M" 1 may be found. Find it and them.
Further information. See MQM Chapter 6 for a summary and
simple matrix manipulations. See Chapter 10 of Margenau and
Murphy (1956) for a moderately complete account. Ay res
(1962) is a good source of accounts of application of matrices
and the way that they can be used to solve a large number of
mathematical and physical problems. Matrices are the basis of
the formulation of quantum mechanics known as "matrix
mechanics and are indispensable for any thorough discussion
of °group theory.
matrix mechanics. The formalism of quantum mechanics
due to Heisenberg is based on the observation that the
position of a particle along a coordinate q and its linear
momentum along that coordinate, p, must obey the rule
qppq = ih, where ti is Planck's constant h divided by 2w.
If one assumes that the observables of position and momentum
molecular orbitals
141
obey this rule, then one obtains quantitative agreement with
all experimental observations. Yet the rule is quite remarkable
in content because it goes counter to all we have been brought
up to believe in classical physics.
In classical physics we may assign a number to the position
(for example, a distance of 4 m from some origin) and a
number to the momentum {for example, 2.5 kg m I* 1 ). The
product of the numbers for the values of these observables,
10 J s, is the same whether we calculate p X q or q X p, and so
the difference qppq is zero. Heisenberg's contribution was to
assert that the difference is not zero, but is equal to the imagin
ary and very small number in. It follows that q and p cannot be
regarded as conventional numbers. Born pointed out to
Heisenberg that the non vanishing of the difference qppq
would hold if the observables q andp were regarded as
"matrices, for in general the product of two matrices depends
on their order. This is the basis of matrix mechanics: instead
of treating observables as ordinary numbers (socalled
cnumbers, the 'c' denoting something classical) they should
be regarded as matrices {socalled qnumbers, 'q' denoting
something quantal) which satisfy the rule of matrix multipli
cation such that qppq = ih. When mechanical calculations are
carried through on this basis one finds excellent agreement with
experiment. Note that the error introduced by using the
wrong rule qppq = 0, and therefore of treating q andp like
ordinary numbers, is only of the order of h; therefore classical
calculations are good enough when inaccuracies of the order of
Planck's constant can be tolerated.
Heisenberg's matrix mechanics preceeded "Schrodinger's
wave mechanics by an insignificantly short time, and very
quickly they were seen to be equivalent mathematical theories
by Schrodinger himself. Today the languages of each formu
lation are used as convenient. The Heisenberg formulation,
dealing as it does with the matrices, or what is equivalent, the
"operators that represent physical observables, is often more
convenient for formal manipulations, and the Schrodinger
formalism (which is more easily visualizable in terms of its
"wavefunction description of the state of a system) is often
used for the actual calculation of the energy levels and states
of complicated systems.
There is a welldefined meaning to the terms Heisenberg
picture and Schrodinger picture (or representations as they are
often too loosely called) of quantum mechanics. The differ
ence between the pictures is in where the timedependence of
the description of a system is taken to lie. In the Heisenberg
picture the timedependence is borne by the operators (or
matrices): the state remains constant but the operators that
extract the physical information change with time. Therefore
we observers are presented with a changing view of the system,
and conclude that it is evolving. In the Schrodinger picture
the operator for the desired information remains unchanging
in time, but the wavefunction squirms around beneath it, and
once again we are presented with a view of the system as it
evolves in time. The difference between the pictures is simply
one of mathematical formulation and is not of physical
significance. There is an intermediate picture standing between
the Heisenberg and Schrodinger viewpoints: this is the
interaction picture, or Dirac picture . In this picture the motion
is divided between the state function and the operator: the
simple motion (often a harmonically varying motion) is carried
by the operator, and the wavefunction carries the extra, com
plicated, but often slow, motion. This picture is very useful in
the formulation of timedependent "perturbation theory.
Further information. An account of quantum mechanics
entirely in terms of matrix mechanics has been provided in a
short book by Green (1965). The original papers are
Heisenberg (1925), Born and Jordan (1925), and Born,
Hensenberg, and Jordan (1926), and English translations have
been published by van der Waerden (1967). Bom's involvement
is nicely illustrated in the collection of correspondence be
tween him and Einstein {Born 1970). For more mathematics
see Dirac (1958), Kemble (1958), Kramers (1964), and
von Neumann (1955). A fourth picture, to complete those of
Schrodinger, Heisenberg, and Dirac, has been described by
Marcus (1970). The mathematics of the first three of these
pictures are well and simply described in §3.2 of Ziman (1969),
§5.4 of Slichter (1963), and Roman (1965).
molecular orbitals. The molecular orbital (MO) method
gives a popular theoretical description of the chemical "bond,
and is an extension of the idea of "atomic orbital to a collec
tion of nuclei. An electron in a molecule may be found in the
vicinity of all the nuclei, and therefore we can regard it as
142
molecular orbitals
E(H)
■£(H)
— S U ^EMO
FIG. M6. The molecular orbital responsible for bonding in H^.
being distributed with varying density over the nuclear frame
work. The °wavefunction of an electron in a molecule contains
information about its distribution, for according to the Born
interpretation the square of the wavef unction at any point
is proportional to the probability of finding the electron there.
Therefore the wavef unction for the electron in the molecule
may be regarded as a function spreading throughout the
nuclear framework, and its square at any point is proportional
to the electron density. This wavef unction is the molecular
orbital.
The distribution of the molecular orbital should account for
the nature of the bond: we should expect a high amplitude of
the orbital, and therefore a high density of the electron, to
appear where our understanding of the chemical "bond shows
such density to be desirable, namely between the nuclei the
electron is attempting to stick together. Think about diatomic
compounds. In a homonuclear bond (a covalent bond between
two identical atoms) we should expect the orbital to spread
equally over the two nuclei; and as the bond becomes increas
ingly polar (in a heteronuclear bond between different atoms)
we should expect the molecular orbital to have increasingly
greater amplitude on one of the nuclei. In the limit of a pure
ionic bond the molecular orbital is wholly localized on one
nucleus.
It is common practice to treat molecular orbitals in the same
way as atomic orbitals are used to discuss the structure of
atoms: the "aufbau process is applied to a set of molecular
orbitals in order to build up the molecular electronic structure.
As an initial example of this basic idea consider the hydrogen
molecule. The molecular orbital responsible for the bonding is
a symmetrical orbital extending over the two nuclei and
having a considerable density in the region between the nuclei
(Fig. M6). Into this orbital we insert one electron, and then
follow it with a second; according to the °Paufi principle the
latter must enter with its spin opposed to the first (Fig. M6),
and no others can be accommodated. Therefore we see that
in a very natural way the molecularorbital theory accounts
for the importance of spin pairing in the formation of a
chemical bond. This idea of the aufbau principle will be
enlarged on when we have discussed the common approach to
the formation and calculation of molecular orbitals.
In principle, it is possible to imagine an extension to the
hydrogen molecule of the calculation of the solution of the
"Schrodinger equation for the "hydrogen atom, and even the
direct calculation of molecular orbitals for a poly nuclear
molecule. This is horribly difficult, and about the only place
where it has been done is in the case of the hydrogen molecule
ion (H\), but even in that apparently simple oneelectron case
the calculation is not at all easy. Since chemists tend to be
interested in molecules more complicated than Hj a scheme of
approximation of the true molecular orbitals has been devised:
this involves first the °BornOppenheimer approximation (of
freezing the nuclei into chosen geometrical arrangements), and
then the application of the method of "linear combination of
atomic orbitals (LCAO). More details will be found under
those entries; for the moment we shall simply discuss the
application of the LCAO approach to the case of the
hydrogen molecule and one or two other simple molecules.
In the LCAO approximation it is supposed that the true,
complicated molecular orbital can be expressed as a sum of
the atomic orbitals on the constituent atoms of the molecule.
This provides a remarkably good approximation because it
reproduces a number of the essential features of the exact
solution. In the case of H 2 the atomic orbitals of principal
importance are thelsorbitals on each nucleus. The molecular
orbital is then expressed as the sum i^i 5 + ^i 5b . This
obviously treats the nuclei equally, and so the electron is
spread among them equally. It also reproduces the significant
accumulation of charge in the internuclear region. This arises
from the wave nature of the two atomic orbitals: one Isorbital
molecular orbitals
143
FIG. M7. Bonding and antibonding orbitals in
may be considered as a standing spherical wave centred on
nucleus a, and the other as a standing spherical wave centred
on nucleus b, (Fig. M7), The two waves overlap significantly
in the internuclear region, and if their amplitudes have the
same sign they interfere constructively and the total amplitude
in the internuclear region is enhanced. It follows that the
electron density in this region is also enhanced. The energy of
the molecule therefore is lowered by virtue of the lowering of
the potential energy of the electrons, which, on this model,
accumulate in the internuclear region and interact with both
nuclei. (I do not want to complicate this description by
invoking the role of kinetic energy and the distortion of the
atomic orbitals themselves. The true source of the binding
energy must be sought in a consideration of the changes in both
potential and kinetic energy, and the structure of the molecular
orbitals must reflect the distortions of the atomic orbitals that
occur when a bond is formed; see the last part of "bond.!
Finally, it is clear that two electrons will form the bond with
maximum stability for, according to the "Pauli principle, only
144
molecular orbitals
two electrons may occupy the bonding orbital, and then they
must have opposed spins.
The method of linear combinations for the construction of
a molecular orbital leads both to bonding and to 'antibonding
orbitals: the latter are formed when the atomic orbitals overlap
with opposite phase (sign of their amplitude), so that destruc
tive interference occurs and electrons are eliminated from the
bonding region between the nuclei (Fig. M7); electrons that
occupy these orbitals tend to drive the bond asunder. A third
electron added to Hj would have to enter the antibonding
orbital, and so the bond would be weakened.
These ideas can be extended very easily to more complex
molecules, and a diatomic molecule of considerable interest
and importance, and to which it is instructive to apply the
method, is oxygen ; . Our kit of parts consists of two nuclei,
sixteen electrons, and, since this is a beginners' kit, one
1sorbital, one 2sorbital, and three 2porbitals on each
nucleus. The structure may be deduced as follows:
1. The tightestbound orbitals are the 1sorbitals; if the two
nuclei are pinned down at the known interatomic distance of
0%, these two orbitals overlap to a negligible extent; therefore
the bonding orbital they give rise to is exceedingly weakly
bonding, and the antibonding orbital is weakly antibonding.
This situation is illustrated in Fig. M8.
2. The 2sorbitals are the next tightest bound, but are much
larger and overlap significantly. Like the orbitals in H 2 they
form a bonding and an antibonding molecular orbital at roughly
die energies marked in Fig. M8.
3. Next we encounter the 2porbitals, and here two quite
distinct possibilities arise: the orbitals may overlap head on, or
broadside on.
(a) In the former case the enhanced electron density is
accumulated in the internuclear region, and we can expect
a strong bond when the orbitals are in phase and a strong
antibond when they are out of phase. We note that the
electron distribution in such an orbital is cylindrically
symmetrical about the internuclear axis (Fig. M8), and so
it is termed a aorbital, or a rjbond (this takes its name
by analogy with the spherically symmetrical sorbital of
atoms).
Atom A Molecule Afl Atom B
FIG. M8, Schematic energy levels and orbitals of homonuclear diatomic
molecules of the first row elements; occupation as for O3.
(b) The other possibility is for the broadside overlap of
two porbitals. The overlap is not particularly extensive,
and it is greatest in regions outside the internuclear axis;
nevertheless, electrons accumulated in these regions can
exert attractive forces on the nuclei and a moderately
strong bond can result {and a moderately effective antibond
if the overlap is destructive). This distribution is a jt orbital,
or irbortd (by analogy with porbital in atoms).
The complete range of orbitals for molecular oxygen con
structed in this way is shown in Fig. M8. Into these recep
tacles we now inject our 1 6 electrons, and play the game
according to the rules of "aufbau .The first tumbles down and
down in energy until it enters 1sff. The next joins it with
opposed spin, the next enters 1s<7*, and so it goes on, and we
encounter no ambiguity until we have inserted 14 electrons. (A
slight hesitation might be noticed at the filling of 2p7T, which
molecular orbitals
145
takes 4 electrons; but we should remember that 2pir is really
two molecular orbitals, one formed from 2p overlap and the
other frorr 2p y overlap. Electron 15 we insert into the antibond
ing 2p7T*arbital. Electron 16 may enter the same orbital with
opposed spin, or it may enter the other 2p3T*orbital of the
pair with either the same or the opposite spin (Pauli allows
either). What determines the outcome? The first Hund rule
informs us that in this situation the lowerenergy arrangement
is that with parallel spins in different orbitals; therefore we
conclude that the "configuration of 2 is 1sa 2 Iscr* 2 2sa 2 2sfJ* 2
2po2 2pn* 20^* 2p n* Is there a way of testing whether
this configuration is plausible? A powerful way is by "electronic
spectroscopy, but another more immediate way is to note that
the presence of the two unpaired spins leads us to predict that
O^ is paramagnetic: see "magnetic properties. It was an early
triumph for molecularorbital theory that Q 2 is in fact paramag
netic.
How does this structure fit in with a more elementary view
that 2 is a molecule with a double bond, 0=0? Looking at
Fig. M8 we can imagine that the bonding due to the
1selectrons ts cancelled by the antibonding nature of the
electrons in 1 so  *; likewise the occupied 2sffand 2sa* cancel
in effect; 2pv is occupied but 2po* empty, and so we notch up
1 on our bonding tally; 2p x 7T and 2p tt are fully occupied, but
2p x t! and 2p ir are both halfoccupied, and so we can cut
another notch, I", total the net bonding can hi.' ascribed to
two net bonds, and this we can signify by 0=0, as in elemen
tary chemistry. It should stimulate profound respect for the
early chemists each time their views on molecular structure
views formed more by introspection than by calculation— are
confirmed by modern quantitative theory. Note too that we
also see why a 'double bond' is less strong than two 'single
bonds'. A single bond is generally a full aorbital plus a full
it orbital, and we have seen that a TTorbital does not have
its extra accumulation in the prime bonding region. Earlier
chemists ascribed this to 'strain': how right they were. A
triple bond, which is also depicted in Fig. M9, is formed from
one O and two Torbitals. The stability of double bonds to
twisting (torsion) can also be understood in MO terms:
rotating a CH 2 group in ethene relative to the other reduces
the overlap between the 2pit orbitals, the rrbond weakens,
and the energy of the molecule rises (Fig. M10).
What orbitals contribute best to the formation of molecular
orbitals? First, they must have the right symmetry: it Is no
good attempting to form a molecular orbital from the sideways
overlap of an sorbital and a porbital, for there is no net
"overlap. (Headon overlap of s and p can, of course, occur.)
That criterion satisfied we then require the orbitals to have
about the same energy (the energymatching criterion) and to
be roughly the same size (to have significant net overlap).
These criteria are explored in the Questions. The extent of
overlap can be increased by permitting "hybridization of the
available atomic orbitals, and the study of hydridization and
the formation of molecular orbitals is the basis of the
lifKjIe bond {scT Or po9
double bond CO4K)
triple
bond
tcr+nfn)
itoined
banana
bonds
FIG. M9. J"he formation, and two
representations of their appearance,
of single, double, and triple bonds.
The bananas (which resemble the
classical picture of strained bonds)
are formed by taking appropriate
sums of the 0, tr representations
of the bonds.
146
molecular orbitals
FIG. M10, Overlap and the torsional rigidity of a double bond.
molecularorbital approach to the discussion of molecular
shape.
All that has gone before suggests that molecularorbital
theory is a modern triumph. Is it? In a word, yes. One can level a
number of criticisms at it, but at the expense of losing the
conceptual simplicity of the theory these can all be overcome.
At a basic level the molecularorbital method underestimates
the tendency of electrons to stay apart: the simple theory of
H 2 , for example, ascribes too much importance to structures
like H H~. This can be repaired by doing some configuration
interaction. The method is also poor at large distances: the
separation of the hydrogen atoms in H 2 yields H* and H" in the
products instead of just 2H; but this too can be overcome by
permitting configuration interaction. At a different level lies
the criticism that the molecularorbital method does not reflect
the chemist's view that different regions of the molecule can
be considered for many chemical purposes as separate, isolated
entities. The proponent of molecularorbital theory can retort
quickly that it is possible to take the molecularorbitals we
have discussed, and which spread throughout the nuclear
framework, and manipulate them into a set of 'localized
orbitals which can be ascribed to different regions of the
molecule. Therefore the criticism can be circumvented by the
application of a mathematical transformation.
The molecularorbital theory can be adapted to quantitative
calculation according to the method of "selfconsistent fields;
an enormous amount of effort has been put into the calcu
lation of accurate molecular wavefunctions and energies by an
extension of the methods used for atoms, and for more
information see "Huckel method and semiempirical methods.
See also °ab initio.
Questions. 1. What is meant by the term 'molecular orbital',
and what information does such an orbital contain? What
features should we expect a molecular orbital to possess:
where is its maximum amplitude expected, and what can we
say about its symmetry? Describe the changes that occur in
the distribution of a molecular orbital as the bond it describes
changes from pure covalent to pure ionic. What is the bonding
energy due to over this range? What commonly used
approximation is resorted to in order to set up a molecular
orbital? What are the deficiencies of this method? Discuss the
formation of the hydrogen molecule in terms of molecular
orbital theory. Account for the instability of the molecule
He 2 on this basis. Set up a molecularorbital scheme like that
for 0^ shown in Fig. M8for all the diatomic molecules of the
first row of the periodic table. Put the following molecules in
order of increasing stability by referring to the diagrams you
have just deduced: Ca, C2, Ci N^, N 2 , N 2 ; Oj, 2 , Oj,*
^2. F?. F2." N&2, Ne 2 , Ne 2 , What is the major defect of a simple
molecularorbital treatment? How may it be overcome?
2. Take the molecular orbital 1s + Is. and show that when
a o
it is occupied by an electron the distribution can be interpreted
in terms of a large proportion of H + tT in the wavef unction
(think about the square of the function). Now consider two
electrons in Is + 15^ and two in Is — Is,., as in the structure
a b a b
of He 2 : what is the electron distribution in this molecule?
Take the orbital Is + 1s. and insert one electron. Write the
a h
"hamiltonian for the molecule, and deduce an expression for the
energy of the molecule in terms of integrals over the wave
functions. Some of these integrals can be identified with the
integrals that occur in the description of the hydrogen atom,
therefore the energy of the molecule can be expressed as the
energy of the atom plus a part that can be ascribed to the
formation of the bond. Analyse the expression in that way.
Now insert two electrons into the orbital and so consider Hj.
Repeat the exercise, using the appropriate hamiltonian, and
attempt to analyse your expression. What you have probably
forgotten to do is to allow for the *antisymmetrization of the
molecularorbital and valencebond: a synopsis
147
electrons: you should write the wavefunction as a "Slater
determinant, and you will discover an extra enjoyable
contribution due to "exchange. Analyse this new result.
3. Consider an orbital $ on atom a and 4 on atom b, with
energies E and E respectively. Allow them to overlap and inter
act and make the approximation that the extent of interaction is
proportional to the amount they overlap. Show by solving
the "secular determinant that, of the two linear combinations
that may be formed, one moves down in energy, and the
other moves up; that the energy change is greater when the
energies E a and f b are similar and when the overlap is greatest;
that the bonding orbital is more localized on the atom with
lowerenergy atomic orbitals. These calculations illustrate the
criteria mentioned in the text, and also show how the polarity
of a bond reflects the relative energies of the contributing
orbitals.
Further information . See MQM Chapter 10 for a discussion
of the molecularorbital method. A simple account of
molecularorbital theory which fills in the details of the present
discussion and gives many applications to chemically important
molecules is given by Coulson in The shape and structure of
molecules (OCS 9) and by Coulson (1961). See also Murrell,
Kettle, and Tedder (1965), Streitweiser (1961), Salem (1966),
Pilar (1968), Slater (1963), and Doggett (1972). Many drawings
of molecular orbitals for numerous molecules will be found in
Jorgensen and Salem (1973). Calculations of molecular orbitals
will be found referred to in °selfconsistent field, 'ab initio,
and "Hiickei method. Accounts of "hybridization, "equivalent
orbitals, 'antibonding, and 'bond extend this discussion. An
alternative account of molecular structure is provided by the
"valence bond theory, which should be referred to, and the two
methods are compared under 'molecularorbital and valence
bond: a synopsis.
molecularorbital and valencebond: a synopsis. The
details of these techniques are given under their separate
headings. In this synopsis an attempt is made to emphasize
their similarities and differences.
1. Both molecularorbital (MO) and valencebond (V8)
theories seek to describe the structure of molecules, their
shape and their energy, and the valence of the atoms that
compose them.
2. Both theories, at least in their simplest interpretation,
achieve their object by leading to an accumulation of electron
density in regions where it is most effective in interacting with
the nuclei. This region is in the vicinity of, or actually in, the
internuclear region. This common interpretation neglects the
contribution to the total energy of the "kinetic energy: in both
the MO theory and the VB theory significant contributions
to both the potential and the kinetic energies may be ascribed
to the distortion of the orbitals of the atoms constituting the
molecule. This additional, but important contribution is often
neglected in an elementary analysis of the theories: we neglect
it here, but refer to sources in Further information.
3. Both theories achieve the object of accumulating electron
density in the internuclear region by recognizing that electrons
cannot be localized on a single atom when that is part of a
molecule. The MO theory says that if an electron can be on
atom a with wavefunction \b (r ), which we abreviate to a (1 ),
and can also be on atom b with wavefunction 6(1), then,
according to the "superposition principle, its actual distribution
must be determined from the wavefunction a(1 ) ±6 (1 ). The
case of two electrons is obtained by dropping two electrons
into this orbital to form ta<1) ± b (1)3 [a(2J + o(2)j . The VB
theory approaches the problem in a different way, and says
that if atoms a and b were well separated the state of the
electrons (one on each atom) would be well described by the
wavefunction a(\) 6(2), because that is the quantummechanical
description of such a situation; it then pretends that the only
difference in the function when the two atoms are at a bonding
separation is that the electron originally on atom a may be on
atom b, and vice versa. According to the "superposition
principle this state of the two electrons is described by the
function a(1) 6(2) ±6(1) a(2). Analysis of the expression for
the energy in both the MO and VB cases leads to the con
clusion that the + sign in the composite functions gives the
lower energy. The implication of this is that the electrons
that form the bonds must enter these wavef unctions with
paired spins (this is required by the "Pauli principle); therefore
both the MO and VB theories account for the importance in
chemical bonding of the electron pair.
148
momentum
4. Although both methods emphasi2e the role of the
electron pair, in practice they do so in different ways. The
MO theory starts by ignoring the way that the electrons enter
the molecule and calculates the molecular orbitals that may
be formed from the available atomic orbitals. At the end of
that work it inserts the electrons in accord with the "aufbau
principle and, perforce, the "Paul! principle. The VB method
concentrates on electron pairs from the outset, and calculates
the energy of various ('canonical') structures that have all the
electrons in the molecule paired in all possible ways. Then
having set up all these 'perfect pairing' structures it allows
them to interact {that is, the true wavef unction is expressed
as a "superposition of them), and then calculates the energy
of the best combination. This process introduces the concept
of resonance.
5. Electrons are allowed to spread over the whole molecule
automatically in the MO method, but these delocalized orbitals
may be transformed into a collection of "localized orbitals. In
VB theory attention is concentrated on individual bonds right
from the beginning, and this feature is largely preserved in the
final superimposed wavef unction. Complete derealization, of
the sort found in 'benzene, has as its counterpart strong
"resonance, as between equivalent Kekule' structures.
6. Although both methods give a similar distribution of
electrons, there are notable differences. Expansion of the
twoelectron MO given in Note 2 leads to
a (1)« {2) + b (1) b (2) ±a (1) b (2\ ±b (Da (2). This
differs from the VB function in the occurrence of the first
two terms. These can be interpreted as the contribution to the
total state of the situation in which both electrons are on the
same atom, either on a or on b. As these extra terms appear
with the same weight as the other terms, we conclude
that the MO theory does not take into account the effect
of electron correlation, the tendency of electrons to keep
apart. There must be some probability of finding both
electrons simultaneously on a or on b, and so we should
expect the true function to be of the form
8(1)6(21 + 6{1)a(2) + Xa(1)a(2) + ju6(D6 (2)
with X, p. < 1 . This modification can be introduced by
the method of configuration interaction. The VB theory
moves to the opposite extreme and forbids both electrons
to be on a or b simultaneously; therefore it overestimates
the role of electron correlation. It can be improved by
adding ionic terms to the original covalent wavefunction
(ioniccovalent "resonance).
7. The MO method is applied quantitatively by
feeding the electrons into approximate wavefu net ions,
and then permitting the orbitals to distort in response
to the electronelectron repulsions. This is taken care of
by doing a self consistent field (SCF) calculation. The
final answer is improved by permitting configuration
interaction, MO theory has received far more attention
than VB theory at this quantitative level because SCF
methods are easily programmed for electronic computers;
the difficulty of dealing with VB calculations has been
the very large number of canonical and ionic structures
that must be taken into account.
Further information. See MOM Chapter 9 for more
details of the methods and their comparison. See
especially Coulson's The shape and structure of molecules
IOCS 9), which is mostly MO, Pauling (I960), which is
mostly VB, and Coulson (1961) which compares them.
A very careful scrutiny of the nature of the chemical
bond has been given by Ruedenberg (1962) and Feinberg,
Ruedenberg, and Mehler (1970). See Murrell, Kettle,
and Tedder ( 1 965) for another comparison of the
methods. See the individual entries on "molecular orbitals
and "valence bond for further information on each.
momentum. In classical mechanics the momentum
plays a fundamental role, and the same is true in
quantum mechanics. In accord with the rules of con
structing quantum mechanics, the momentum, an
observable, must be interpreted as an "operator. Once the form
of this operator is known, other observables that depend on the
momentum may also be expressed as operators, and so a com
plete scheme may be formed. The choice of the operator for
linear momentum is of crucial importance in quantum theory,
and one common and familiar choice for the component of
linear momentum along the ^axis is the differential operator
(h/i) (9/9g). It follows from this that the linear momentum of
a system is related to the gradient of the "wavefunction that
multiplicity
149
describes its state: steep gradients correspond to high momenta.
This aspect of the wavefunction is compatible with the
°de Broglie relation, which states that the wavelength of a
wavefunction diminishes as the momentum of the particle
increases {p = A/A). When the system is described by a standing
wave the average gradient is zero, and in such states it follows
that the linear momentum is zero. For example, a "particle
trapped in a onedimensional square well is described by a
standing wave, and its mean momentum is zero. {Classically
that would be interpreted as multiple reflections from the walls
of the container reversing the momentum so often that its
mean vanished.)
If a particle's linear momentum along an axis is sharply
defined, its position on the axis is indeterminate: this is an
important consequence of the 'uncertainty principle and is an
aspect of the waveparticle duality of matter. It can be
understood by recognizing that a particle with definite
momentum is described by a monochromatic wave of indefi
nite extent, and in such a wave, according to the Born interpre
tation (see "wavefunction), the position of the particle occurs
with equal probability throughout space. Conversely, the
formation of a "wave packet, which localizes the position of
the particle, does so at the expense of superimposing so many
waves of different wavelength that the momentum is broadly
dispersed.
Questions. 1. What aspect of the wavefunction determines the
momentum of a state? How may the de Brogtie relation be
justified in terms of this interpretation? Under what circum
stances is the momentum of a particle zero? What is the
momentum of a particle trapped in the ground state of a
onedimensional square well? Why may the "kinetic
energy be nonzero even though the linear momentum is
zero? In what sense are the linear momentum and the
position of a particle "complementary?
2. By average value of the gradient is meant the "expectation
value. Evaluate the expectation value of the linear momentum
for particles described by running waves of the form exp ikx
and exp(— \kx). (Note that complexconjugate wavefunctions
correspond to apposite momenta.) Calculate the expectation
value for the linear momentum of a particle in a one
dimensional squarewell potential (see "particle in a square
well for the wavefunction). Prove from the hermiticity of
the linear momentum operator that the expectation value of
the momentum for a state described by a real wavefunction
is necessarily zero.
Further information. See MQM Chapters 3 and 4 for a detailed
discussion of momentum in quantum mechanics. The
fundamental role of linear momentum in quantum theory is
described in Bohm (1951), Messiah (1961), Schiff (1968),
Landau and Lifshitz (1958a), Dtrac (1958), von Neumann
(1955), and Jauch (1968).
multiplicity. The multiplicity of a "term is the number of
"levels it possesses; that is, it is the number of different values
of the 'quantum number 'J that may be ascribed to the term.
When the values of the quantum numbers L and 5 of the
term are such that L > S the multiplicity is equal to 2S + 1.
This is because that number of values of the total angular
momentum J may be formed by coupling the spin and
orbital "angular momenta together {J may take the values
L + S, L + S  1  L  S ). When L < S the number of
lvalues that may be formed is 2£ 4 1, and so under these
circumstances the multiplicity is equal to 2i + 1.
The numerical value of 2S + 1 is normally denoted by an
upper left superscript on the term symbol, but it is important
to note that this gives the true multiplicity of the level only
when L > S. As an example a 2 D term ('doublet D') has two
levels distinguished by J =  and J m j and written ^D^ and
2 D 5 / 2 ; likewise a 3 P term (a triplet P term) has a multiplicity
of three, and its levels are distinguished as 3 P 2 , 3 P[, 3 P . The
^S term, with L = and S = \ is referred to as a doublet
term even though it has only one level {J = \). Beware of
sloppy usage, and always think about the relative size of
Sand L.
Questions. What is meant by the multiplicity of a term? How
can it be calculated from a knowledge of the values of L and S?
What is the significance of the left superscript on a term
symbol? Under what circumstances does the superscript
indicate the multiplicity? How many levels do the following
terms possess (that is, what is their multiplicity):
150
multiplicity
2 P, 3 P, *P, l S, S S, 3 D? In each case indicate the J labels
of the levels.
Further information . The multiplicity of a term appears
spectroscopically as the "fine structure. For the structural
differences of singlet and triplet terms see the entry singlet
and triplet states. For general aspects see angular momentum.
For a further discussion see MQM Chapter 8 and books on
atomic and molecular spectroscopy: Whiffen (1972),
King (1964), Herzberg (1944), Kuhn (1962), White (1934),
Candler (1964), and Condon and Shortley (1963}.
Stevenson (1965) gives a moderately simple and complete
theoretical account of the multiplet structure of atoms and
molecules. See Calvert and Pitts (1966) and Wayne (1970)
for an account of the way that the multiplicity of a species
determines its chemical behaviour.
N
node. A node is the place where a "wavefunction has zero
amplitude (that is, no displacement). The node may be a point,
a line, or a surface. For the 'particle in a one dimensional
square well the wavefunction has a node at the walls and at a
number of regularly spaced points within the box, and the higher
the energy (or higher the harmonic of the fundamental wave)
the more nodes are present. A 1 sorbital in 'hydrogen has no
nodes, apart from a rather special one at infinity (see 'atomic
orbital). A 2sorbital has one node which should be visualized
as a spherical surface surrounding the nucleus; as the position
of this node depends only on the radius and is spherically
symmetrical it is called a radial node. A 2porbital has no
radial node (apart from the one at infinity), but it is divided
into two lobes by an angular node, which is a plane running
through the nucleus. A more complicated nodal structure
occurs in the other "atomic orbitals, but an sorbital always
has no angular nodes, a porbital always has one, and a
dorbital always has two.
The significance of the nodal structure of a wavef unction
stems from its connexion with the "momentum or the
'kinetic energy of the system: the more nodes in a given
region the greater the kinetic energy. The reason for this is as
follows. A node occurs where a wavefunction changes sign,
and the number of sign changes in a region increases as the
wavelength shortens. Therefore as a shorter wavelength
implies, through the °de Broglte relation, a greater momentum,
it follows that the more nodes present the greater the mo
mentum and the kinetic energy. The connexion is illustrated
by the example already mentioned of a "particle in a square well.
In the case of angular nodes the relevant momentum is the
•orbital angular momentum; so we can believe that as the
number of angular nodes increases so too does the angular
momentum. This is confirmed by calculation, for the number
of angular nodes is equal to the numerical value of the
'angular momentum quantum number B, and the magnitude of
the angular momentum is proportional to J [K(fi + 1 )] . For
this reason the angular momentum of a delectron (2 angular
nodes) exceeds that of a pelectron (1 angular node).
Questions. What is a node? What shape can it take? Can there
be a nodal point (rather than a nodal line) for a particle in a
twodimensional welt? How are the nodes in an atomic
system classified? How many nodes are there in the wave
function of atomic hydrogen corresponding to the principal
quantum number n? Does the number of nodes depend on n
and £? Does the value of m% affect your conclusions? Why is
there a connexion between the number of angular nodes and
the orbital angular momentum of a state? Discuss the nodal
structure and the physical significance of the nodal structure
of the orbitals in diatomic molecules.
Further information. See MQM Chapter 3 for a variety of
different systems showing nodes of various kinds. The nodes
of functions, being places where the functions drop to zero,
may be ascertained by locating the zeros of the function; the
zeros of many mathematical functions are listed in Abramowitz
and Stegun (1965). A general theorem on nodes states that the
lowest energy level is nodeless: see §18 of Landau and Lifshitz
(1958a). This theorem fails when many particles are present
because Fermi holes occur in the wavefunction by virtue of
"spin correlation.
151
152
noncrossing rule
noncrossing rule. Consider two states of an atom or mole
cule, and let their energy depend on some parameter P (for
example, a bond length}. As this parameter is varied the
energies change, and it is conceivable that a variation of P
takes the energy of the upper energy state below that of the
lower; that is, the energy curves cross. The noncrossing rule
asserts that this crossing cannot occur if the states have the
same symmetry. It follows that a variation of P leads to the
energy variation illustrated in Fig. N1.
FIG. N1. The noncrossing rule. Black lines correspond to states of
different symmetry (which mav cross) and colour lines to states of the
same symmetry (which may not).
The rule is of considerable importance in the construction
of correlation diagrams (see "united atom), because its
application enables the energy of states of molecules and atoms
to be followed as bonds are formed.
Note that the noncrossing rule is an example of an
adiabatic process: if a system starts in \j/ 2 (Fig. N1 ) and the
parameter is varied slowly, it will be in state t^ ( of the new
system (corresponding to a different value of the parameter/ 3
on the right of the diagram). If the motion from left to right is
done very rapidly, the system may be in \p 2 at the final value
of P: this corresponds to an excited state of the new system.
Therefore we see that the noncrossing rule is applicable to
timedependent systems only if their motions are slow.
Further information, A diagram like Fig. N1 may be deduced
from perturbation theory by solving the Schrodinger equation
for a twostate system. This is done in Chapter 7 of MQM.
Such an analysis formed the basis of Teller's deduction of the
rule (Teller 1937). A more abstract and earlier deduction is
that of von Neumann and Wigner {1929). Both approaches
have been strongly criticized, and a more rigorous proof has
been given by Naqvi and Byers Brown (1972), and extended
to polyatomics by Naqvi (1972). For simpler and more con
ventional accounts, see Couison (1961) and Herzberg (1950).
normal modes. The number of modes of vibration of a mole
cule containing N atoms is 3/V6 in general, but 3/V5 if the
molecule is linear. The source of these numbers is the fact that
to specify the position of an atom requires three coordinates,
and so the specification of the position of all the N atoms in a
molecule requires 3/V coordinates. Changing any of the 3/V
coordinates corresponds to changing the molecule's shape
(bending or stretching its bonds) or to moving or rotating it as
a whole. Of the 3/V coordinates, 3 may be chosen to be the
position of the centre of mass of the molecule (that is, they
specify the position of the molecule in the room), and so the
remaining 3AV3 must specify the position of the atoms with
respect to the position of the centre of mass. Of these, three
may be ascribed to the orientation of the molecule (if it is non
linear), and the remaining 3/V6 must then specify the
relative positions within the molecule of all the atoms, and
changing them corresponds to a bending or stretching of the
bonds, that is, to molecular vibrations. If the molecule is linear
only two coordinates are required to specify its orientation,
and so the number of internal coordinates is 3/V— 5.
Consider now the case of carbon dioxide, a linear triatomic
molecule having 9 — 5 = 4 internal degrees of freedom
(coordinates necessary to specify the configuration of the
molecule other than its position or orientation in the world).
Suppose we identify one of the degrees of freedom with the
stretching of one of the C— O bonds, and investigate the vi
bration of that bond. When the vibration of the bond is
excited the other equivalent C— O bond will very quickly pick
up its energy, because of the motion of the shared carbon
atom, and there will be a "resonant transfer of the vibration
from one bond to the other. The process will continue, and
the vibrational motion transfers back and forth between the
bonds until some external process quenches the molecular
normalized function
153
villi at ion. Suppose now that instead of exciting just one of the
bonds we were more canny and excited both equally, we
should expect an equilibrium situation in which the vibration
continued smoothly until it was quenched. Take, for example,
the excitation of the symmetrical stretching mode
O — C — K) * — >C< — 0: the carbon atom is buffeted
equally from both sides, and there is no way (unless
"anharmonicities are present) for the alternative combination,
the antisymmetries! stretching mode *— 0— C+K3 ^
0**C— CH>, to be excited. Nor can energy in these modes be
transferred to the bending modes, because to do so a perpen
dicular force is needed (this can be provided by the 'Coriolis
interaction arising from molecular rotation). Thus we see that
by a judicious choice of the modes of vibration of the molecule
we may obtain a set of independent motions: these are the
normal modes. The four internal (vibrational) modes of C0 2
may be chosen as four normal (independent) modes, and
because only four internal modes exist, any vibration of the
molecule, however complicated, may be expressed as a super
position of the normal modes.
The independence of the normal modes makes their
quantummechanical discussion very simple, since each of
FIG. N2. Normal coordinates in C0 2 and H 2 (or other triatamic
molecules).
them may be considered to be equivalent to a single "harmonic
oscillator of a particular mass and 'forceconstant vibrating
along some equivalent coordinate, the normal coordinate. All
the properties of "harmonic oscillators apply in the normal
way to each normal mode, and the "selection rule that governs
a vibrational transition applies to the modes too. In particular,
a mode is active in the infrared region of the spectrum (and so
gives a line in the ordinary "vibrational spectrum) only if there
is a change of 'dipole moment along the normal coordinate;
that is, if the dipole moment of the molecule changes when it
is distorted in a particular normal mode. Some of the normal
modes of carbon dioxide and water are illustrated in Fig. N2.
Questions. Why does a nonlinear molecule possess 3N— 6
vibrational modes and a linear molecule 3/V5? What happens
to the extra mode in the linear case when it is turned into a
bent molecule? What happens when a single bond is stretched
and then released in C0 2 , H 3 0, and CH 4 ? How is it possible to
achieve a 'steadystate' vibration? Under what circumstances
does the independence of the normal modes fail? What
advantages stem from the use of normal modes? What is the
basis of the quantummechanical calculation of the vibrational
frequencies of the normal modes? What factors affect their
frequency? If 12 C is replaced by 13 C in COj, would you expect
all the normalmode frequencies to change?
Further information. See Chapter 1 of MQM For a further
discussion of the deduction and properties of normal modes
and vibrations. See Chapter 6 of Brand and Speakman (1960)
for a simple discussion of the classical and quantum mechanics
of normal vibrations. A helpful introduction to normalmode
analysis is given by Woodward, (1972), and more advanced
sequels may be found in Wilson, Decius, and Cross (1955). See
also Barrow (1962) and Ganz (1971). All these books describe
the "grouptheoretica! analysis of normal modes of vibration.
normalized function. The Born interpretation of the wave
function views \p*ir)\p{t)dT as proportional to the prob
ability of finding the particle in the volume element dr
surrounding the point t; when the proportionality constant is
unity the wavefunction is said to be normalized (or, more
strictly, normalized to unity).
154
nuclear magnetic resonance: a synopsis
Suppose the wavef unction we are presented with is ^;
how do we proceed to normalize it? The basis of the method is
the observation that, if the wavef unction is to be related to
the probability distribution of the particle, then the prob
ability of it being somewhere in the universe must be exactly
unity. The probability of the particle being in the universe is
the sum of the probabilities of it being in each of the vofume
elements dr into which the universe may be shattered; this total
probability is the integral /dr^*^, which must equal unity.
Therefore if in fact we find it equal to N we may normalize
the wavef unction by dividing it by the number N y ' : the
normalized function is N~ y ' \p and N~* is the normalization
constant When a function is both normal and 'orthogonal it
is orthonormal.
Questions. What is meant by the term 'normalized function'?
Why is it convenient to deal with normalized functions in
quantum mechanics? What is the procedure for normalizing an
arbitrary function? Normalize to unity the following wave
functions: the constant a in a universe stretching from x =
to x = L; the functions sin*x in the same short universe; the
function expim0 in a universe stretching around a circle; the
function exp(— r/a ) in the whole of threedimensional space.
Further information, Simple examples of normalizing wave
functions are given in Chapter 3 of MQM. A particular
problem arises with the normalization of functions that do
not decay at large distances; for example, the function expixx
oscillates for ever as x — ♦ •». Resort is then made to a special
device (5function normalization): this is described in §5 of
Davydov (1965), §4 of Mandl (1957), and §1.3 of Goldberger
and Watson (1964).
nuclear magnetic resonance: a synopsis. The technique
of nuclear magnetic resonance (n.m.r.) is the observation of
the absorption of electromagnetic radiation by the magnetic
nuclei of molecules, and in particular of protons, in the
presence of an externally applied magnetic field.
In a magnetic field the two orientations of the 'spin of a
proton have different energy (a lies below j$) and transitions
between them (inversion of the orientation, or reversal of the
direction of spin) can be induced by an electromagnetic field
of the appropriate 'resonance frequency. Typical spectro
meters employ a 15 kG magnetic field, and the resulting
energy separation corresponds to photons of frequency
60 MHz; recent developments have taken magnetic fields into
the 50 kG region, where the radiation required is about
200 MHz. These figures imply that n.m.r. is a form of radio
frequency spectroscopy. The experiment is generally performed
by applying a fixed radiofrequency field to the sample, and
varying the applied magnetic field until the radiation is
absorbed most strongly: this is the resonance condition. A
typical spectrum is shown in Fig. N3.
k;
dtemieo t shftV
H
PV
FIG, N3. The structure of an n.m.r. spectrum (of acetaldehydel. In
each case the signal is due to resonance of the starred proton.
The information forthcoming from an n.m.r. spectrum is as
follows:
1 . The position of the resonance. Protons in different
chemical environments resonate at different values of the
applied magnetic field: this is the 'chemical shift. It arises
nuclear statistics
155
because the applied field may induce local fields in the mole
cule, and the nuclei sense the total field. The applied field
induces different fields in different types of chemical groups,
and so different groups resonate with the fixed radiofrequency
at different values of the applied field. More information will
be found under ^chemical shift. The simplest application of
the chemical shift is to the recognition of different types of
groups in an unknown molecule (see Fig. N3).
2. The fine structure of the spectrum. Under high resolution
the lines of a spectrum are normally found to have a fine struc
ture. This is due to 'spinspin coupling, in which the spins of
the magnetic nuclei interact. As a first approximation the
effect of the presence of the magnetic moments of the other
nuclei is to modify the local magnetic field at the nucleus of
interest, and the consequence of this is that it resonates at a
value of the applied field which depends on the orientation of
the neighbouring nuclear spins. The spinspin coupling within
a magnetically equivalent group of nuclei (a group with the
same chemical shift) may be large, but it does not appear in
the spectrum (this is a consequence of the selection rules that
govern the spectrum). When the magnitudes of the spin spin
coupling and the .chemical shift are comparable the spectrum
may take on a very complicated appearance, but when they
are markedly different the interpretation of the spectrum is
simple (Fig. N3). The fine structure is an excellent fingerprint
for the identification of an unknown molecule, or for the
determination of structure.
3. The width of the tines; the line shape. The shape of the
lines, and especially their width, is determined in solution by
relaxation processes: these are described under the heading
"electron spin resonance. The same effects operate in n.m.r.,
but as the nuclear magnetic moment is about 2000 times
smaller than the electron spin magnetic moment its interaction
with the environment is very much weaker and the relaxation
times correspondingly longer (and line widths much less).
Nevertheless, the determination of line widths and relaxation
times is an important tool for the study of molecular motion
in fluid solution. It is also important for the study of chemical
motions; for example, tautomerism and proton exchange. The
line shape is strongly affected by processes that occur on the
n.m.r. time scale: if a motion modulates the chemical shift of
a nucleus (for example, if a proton jumps between two
inequivalent environments) then the line will be broadest
when the frequency of the motion is of the order of the
frequency difference between the two resonant positions.
Further information. See the entries on the chemical shift
and °$pinspin coupling for more information, and an idea of
the magnitudes involved. The magnetic moments of nuclei are
listed in Table 17. A description of n.m.r. will be found in
McLauchlan's Magnetic resonance (OCS 1): this gives a des
cription of thu method and the way that ;t may be applied
For other simple accounts see LyndenBell and Harris (1969),
Jackman (1959), Roberts (1959), Carrington and McLachlan
(1967). More details will be found in Pople, Schneider, and
Bernstein (1959), Emsley, Feeney, and Sutcliffe (1965),
Stichter (1963), and Abragam (1961). Recent advances are
described in Advances in magnetic resonance, Progress in
nuclear magnetic resonance spectroscopy, Annual review of
n.m.r. spectroscopy, and in the Specialist periodical reports of
the Chemical Society. For a description of rate processes in
terms of n.m.r. see the above books, especially Chapter 12 of
Carrington and McLachlan (1967).
nuclear Statistics. Nuclei, like electrons and other particles,
must satisfy the requirements of the "Paul! principle: whenever
any two equivalent nuclei are interchanged the overall wave
function must change sign if they are "fermions, but not change
sign if they are "bosons. This requirement has stringent con
sequences on the possible "rotational energy levels that a mole
cule may occupy, for the rotation of a molecule is a mode of
motion that interchanges nuclei. Unfortunately, rotating a
molecule also drags round the electrons, and so it is necessary
to disentangle the exchange of the nuclei from the other effects
that accompany rotation. We shall confine our attention to the
hydrogen molecule, partly because it is pleasantly simple, and
partly because it is Important through the role that nuclear
symmetry plays in the thermal properties of hydrogen gas.
Consider what happens when a molecule of hydrogen is
rotated through 180°. The nuclei are interchanged, but so too
is the orientation of the molecule as a whole, and the electrons.
Inspection of Fig. N4 shows that it is a simple matter to
return the electrons to their initial position in space by in
156
nuclear statistics
FIG, N4, Symmetry operations on nuclei and electrons in a
hornonuclear diatomic.
verting them through the centre of the molecule (no sign
change on the electronic wavef unction for Hj in its ground
state), and then reflecting them back across the plane in
which the molecule was rotated (also no sign change in the
case of the ground state of Hj ). A glance at the figure shows
that the object of interchanging the nuclei without any net
effect on the electrons has been achieved.
At this point we analyse the effect of each of the con
stituent operators on the wavef unction of the molecule.
The wavefunction is the product of the wavefunctions for
the electronic state of the molecule \b „ the vibrational state
y/ ... the rotational state \b , and the nuclear state J/
(of which more below). When the molecule is rotated through
180 the rotational wavef unction changes sign if the
"rotational quantum number,/ is odd, but does not change
sign if it is even (think of the similar behaviour of s, p, d, f,
. . . orbitals). Therefore from the overall rotation of the mole
cule, the first step of the chain, we get a factor of (—1 ) J . On
the next two steps, the inversion and reflection of the electrons
only, there is no change in the case of Hj (although molecules
in states other than E may change sign), and so at stage D the
wavefunction has changed sign by (1) J . Finally, the spin
orientations of the nuclei are interchanged: this takes the
molecule from D to E, and the last stage results in the same
molecule that would have been obtained simply by relabelling
the two nuclei.
When the two protons are relabelled the Pauli principle
demands that the total wavefunction change sign; therefore
going from A to E changes the sign of the function, and so the
chain of operations A— B— C— D— E must also lead to a change
of sign. So far a factor of (— 1) has been identified, but
the step D to E may also lead to a sign change. This may be
appreciated by considering the possible spin states of the two
equivalent protons: the spins may be parallel or ant i parallel,
just as in the case of two electrons. The wavefunction for the
spins in their anti parallel configuration is a n (a)p\ (b) —
$ n (alcc^b) (see "singlet and triplet states), and if the projections
a n and j3 n are interchanged this function changes sign. Con
versely, the parallel spin arrangement (a nuclear triplet) is
represented by the three statesa (a)a (b), a (a)p 5 (b) +
/J lata (b), (alp^ (b), and none of these changes sign when
the orientations are exchanged.
Suppose that the protons of H a are antiparallel, then the
step from D to E introduces a factor of (—1) into the circuit,
and so the overall phase in the trip from A to E is (—1 ) + .
But as the protons are fermions the wavefunction must change
sign on going from A to E, and so we are forced to the con
clusion that when the nuclear spins are antiparallel the mole
cule can occupy only the rotation states corresponding to even
values of the quantum number J. Conversely, if the proton
spins are parallel the step from D to E leaves the sign
unchanged, and so overall a factor of (—1 ) is obtained for the
cycle from A to E; it follows that when the proton spins are
parallel the molecule can occupy only the states with odd
values of J.
The restrictions on the quantum numbers of the occupiable
states have two principal effects. The first is the modification of
the appearance of the spectrum of the molecule: in a thermal
equilibrium mixture the oddJ states are occupied three times
as heavily as the states with even J because three orientations
of the nuclear spins are compatible with odd values of J, and
only one orientation with even values. This intensity altern
ation is characteristic of the rotational structure of the spectra
nuclear statistics
157
of molecules containing equivalent nuclei. Hydrogen molecules
with paired nuclear spins (and therefore even J values} con
stitute parahydrogen, and those with parallel spins (and
therefore oddJ values) constitute orthohydrogen. At thermal
equilibrium a sample of gas contains three times as much
ortftohydrogen as parahydrogen, except at the lowest
temperatures, where all the gas tends to occupy the lowest
rotational level J = 0, in which case the thermalequilibrium
sample contains only parahydrogen, Ortho and para
hydrogen display different thermal properties at low
temperature because of the differences in their available
rotational energy levels.
The attainment of thermal equilibrium can be a very slow
process because the relative orientation of the nuclear spins
has to be changed. A pure parahydrogen sample will change
only slowly at room temperature to the thermalequilibrium
mixture, because many of the nuclear spins must be reorien
tated to make them parallel to their partners. This can be
achieved more rapidly in the presence of a catalytic surface on
which the molecules may dissociate and then recombine with
random partners, or in the presence of a paramagnetic molecule
(such as oxygen or a transitionmetal ion) in which the
magnetic field of the molecule can interact more strongly with
one proton than the other and so drive them into new relative
orientations (see Fig, S7 on p. 218).
The nuclear statistical effects are lost when the molecule is
isotopically substituted, for then the nuclei are no longer
equivalent. The molecule HD shows none of the properties of
the kind just described. On going to D 2 a situation with
equivalent nuclei is regained, but the deuterons are bosons and
the intensity distribution is modified accordingly (see
Questions). The effect of nuclear statistics on spectra and
thermal properties may be discovered also in other molecules
containing equivalent nuclei, such as 2 , HCCH, H 2 0, CH 3 CH 3 ,
CH 4 ; the analysis gets quite complicated but depends on the
same arguments.
Questions. 1 . What is the basis of the effect of relative nuclear
orientation on the rotational energy levels of molecules? Do
the nuclei affect the energy levels or just their population?
Why is it necessary to consider the peculiar complicated
scheme depicted in Fig. N4? Why are evenJ states associated
with paired nuclear spins when the nuclei are fermions, and
odd./ states associated with parallel nuclear spins? What
effects does the demand of nuclear statistics have on the
spectral and thermal properties of hydrogen? What is the
significance of the classication of hydrogen into ortho and para
states? What is the thermalequilibrium concentration of the
two species at elevated temperatures? Why is the interconirersion
a slow process, and what can be used to accelerate it?
2. Consider the molecule D 2 , the spin of the deuteron being 1,
and determine which nuclear states may be associated with
even and odd J values. What is the lowest state of the molecule?
Show that at elevated temperatures the even:odd J state ratio
is 2:1.
3. Deduce that the ratio of even:odd nuclear spin states for a
nucleus of spin / in a molecule containing two equivalent
nuclei of spin / is (/ + 1):/. Proceed by counting the number
of odd and even combinations of the states Im Im that may
a b '
be formed.
Further information. See MOM Chapter 10 for a further dis
cussion and a deduction of the general rule. A simple dis
cussion will be found in §4.5 of Sugden and Kenney (1965),
and §5.13 of King (1964), who also discuss the spectral
consequences. The thermal consequences are described by
Davidson (1962). See Townes and Schawlow (1955) for the
extension of these arguments to more complex molecules.
Operators. Classical mechanics deals with observables such as
position and momentum as functions, sometimes of each
other, or of time, and Newton's laws of motion enable these
functions to be discovered. Quantum mechanics recognizes
that all the information about the system is contained in its
"wavefunction and that, in order to extract the information
about the value of an observable, some mathematical operation
must be done on the function. (This is analogous to the
necessity of doing an act, an experiment, on the system in
order to make a measurement of its state.} Quantum mechanics
really boils down to making the correct selection of the
operation appropriate to the observable.
In the simple quantum mechanics that concerns us it turns
out that the right way to determine the momentum from a
wavefunction is simply to differentiate it and then multiply
the result by h/i. Thus the gradient of the wavefunction at a
particular point determines the "momentum. The operator
that extracts the position turns out to be simply 'multipli
cation by x', but this, as you can imagine, is deceptively simple.
Once we know what the operators are for the dynamical
variables of position and momentum we can set up the
operators for all observables, because these can be expressed
as functions of the two basic variables. Thus the kinetic
energy in classical mechanics is a function of the momentum,
namely p 2 /2m, withp 2 = p 2 + p 1 + p 2 ; therefore the
corresponding operator can be obtained by replacing p 1 by
(h/i} 2 (3/3x) 2 , etc.; this shows that the curvature of the wave
function determines the "kinetic energy.
How does one find the operators forp and x in the first
place? The choice is severely limited by the requirement that
the operators be such that the values of the observables they
yield are real numbers (the result of an observation cannot be
a complex number}; this implies that the operators must have
the mathematical property of hermiticity (they must be
hermitian operators). Another requirement is that the
operators must satisfy the rule that (xp —p x)\}/ must be
equal to ih 4> (We have denoted the operators corresponding to
the observables x and p x by x andp^,) Another way of putting
this is that the "Commutator of x andp must be ih (see
"matrix mechanics). The latter is a very stringent requirement
and has profound consequences; from it one may deduce the
"uncertainty principle.
Having found the operator for the observable of interest,
the value of the observable for the state of the system in
question is an "eigenstate of the operator, if the state is not an
eigenstate the result of the experiment is determined by the
'expectation value of the operator.
Questions. I, Why are operators important in quantum
mechanics? What is the operator corresponding to linear
momentum in the xdirection, and in the /direction? What is
the operator corresponding to the position along the z
coordinate? What is the operator corresponding to kinetic
energy, and to thezcomponent of angular momentum? The
state of a system is described by the function exp \kx: what is
the linear momentum of the state, and what is its kinetic
energy? Another system is described by the function coskx:
what is its linear momentum and kinetic energy?
2. What properties must operators possess if they are to be
satisfactory in quantum mechanics? Confirm that the
158
orbital angular momentum
159
"commutator of x and (h/i){d/dx) is ih. If we had chosen
'multiplication byp^' to be the operator corresponding to the
xcomponent of linear momentum, what would have been the
necessary choice of operator for position? In the momentum
representation of the operators just encountered, what would
be the appropriate expression for the kinetic energy, and the
Coulomb potential energy of two charges at a separation r?
Further information. Operators are at the very heart of
quantum theory, and so books dealing with the fundamentals
treat operators at length. For a simple account of the basic
theory see MQM Chapter 4. The classical account of operators
and observabfes is provided by Dirac (1958), and more
mathematical accounts will be found in Mackey (1963),
von Neumann (1955), Jordan (1969), and Jauch (1968). For
a resume, see Appendix 1 of Roman (1965). An introductory
account of representation theory is given in Chapter 4 of
Davydov (1965). The formulation of quantum mechanics as
"matrix mechanics uses the properties of operators directly,
and a succinct account is given by Green (1965).
orbital angular momentum. The orbital angular momentum
is the contribution to the total angular momentum that in
classical mechanics would be ascribed to the circular motion of
a particle around a fixed centre. In quantum mechanics it is
found that the orbital angular momentum is quantized, and its
values are constrained in two ways.
1. The magnitude of the orbital angular momentum is
confined to discrete values given by the expression
\\J [£(£ 4 1)] , where E is the orbital angular momentum
quantum number, or azimuthal quantum number, and is
limited to positive integral values (£ = 0, 1, 2, . . . ). Thus the
angular momentum of any body is confined to the values
0, ft/ 2, ftV6, . . .. (Massive rotating bodies, such as a
bicycle wheel, have angular momenta corresponding to
£~ 10 .) In some situations (such as the "hydrogen atom)
the maximum value of 9. is limited by the value of other
"quantum numbers.
2, The orientation of the direction of rotation is quantized
(this is space quantization). The orientation of the plane of
rotation is determined by the magnetic quantum number m%
length VjCj4l)'
classical trajectory
FIG. 01. (a) shows how a vector of length proportional to [K(.t + 1 )] li
is related to a classical trajectory; the component on the raxis is mh In
(b) is illustrated the discrete orientations (with respect to the z'axis)
allowed by quantum theory to a particle with fi • 2.
which can take all integral values between +£ and — fi (there
are [2% + 1] such values). The plane of rotation is determined
in the sense that the component of the orbital angular
momentum about a selected axis (conventionally the^axis)
is limited to the value moh (Fig. 01). Therefore if the
magnitude of the angular momentum of a particular body is
fu/"6 (so that 9. = 2) the angular momentum about the ^axis
may have one of the give values — 2Ji, —16, 0, h, 2h (the different
160
orbital angular momentum
signs correspond to different (classical) senses of rotation)
{Fig. 01b).
The component of momentum about either of the other
axes Uf or y) is indeterminate (according to the 'uncertainty
principle), and so if one denotes the angular momentum of a
body by a vector I of length / [£(£ + 1 )] its projection on only
one axis (thezaxis, by convention) may be determined at
a particular instant: the simultaneous determination of either
the x or the /component is excluded by the uncertainty
principle. Therefore, if we still wish to represent angular
momentum by a vector we have to draw it in a way that does
not give the impression that we know more about its orien
tation than is permitted by the uncertainty principle. The best
we can do is to draw a cone of all the possible (but indeter
minate) positions of the vector (Fig. 01 b), all of them of
length proportional ton/ [£(£ + 1)] and zcomponents pro
portional to /ugh (see ^vector model). Sometimes the cone of
possible positions is interpreted in terms of the 'precession of
the angular momentum; but see the appropriate entry before
you believe in such a description.
The orbital angular momentum of a system is related to the
number of "nodes in its wavefunction: the total number of
angular nodes is equal to £, and the number of angular nodes
that one encounters on encircling the zaxis is equal to Imp I.
Thus a dorbital has two angular nodes, and S.  2; the d 
' xy
orbital has both nodes in the x^plane, and so one encounters
both on a circuit about the zaxis, consequently Imp I = 2.
This connexion between angular momentum and nodal
structure is easy to understand if one recalls the °de Broglie
relation or, what is equivalent, recalls that the "kinetic energy
of a particle increases as the curvature of the wavefunction
increases. Thus a shorter wave length wave (a more buckled
function) has more nodes in a given length than one of longer
wavelength. For angular momentum we are concerned with
momentum on a circle or sphere; therefore a constant
function (that is, a function independent of the angles and <p)
has no nodes, is of infinite wavelength, has zero kinetic energy,
and therefore zero (angular) momentum (or by the de Broglie
relation has zero momentum). A function with one angular
node (such a node lies on a diameter, and cuts a circle twice)
corresponds to one wavelength wrapped round a circle, and
one with two angular nodes (four nodal points on a circle)
corresponds to two wavelengths confined to the same circum
ference, and so on. Therefore we see that the wavelength is
shortened and the angular momentum is increased as we pack
more nodes into the function. This picture of fitting waves on
to a circle also makes clear the reason why angular momentum
is quantized: only integral numbers of wavelengths can be
fitted, for otherwise there would be destructive interference
between waves on successive cycles of the ring (Fig. 02).
FIG. 02. (a) Acceptable end (b) unacceptable waves on a ring.
Questions. What is the classical definition of orbital angular
momentum? How does quantization modify our view of the
classical case? What values of the magnitude of the angular
momentum of a rigid body are permissible? What orientations
of a rotating body are permissible? Why is it not possible to
represent an angular momentum by a vector in a fixed orien
tation? What is the connexion between the number of nodes
in a wavefunction and its angular momentum (and which nodes
do we count)? The number of nodes around the equator deter
mines Imgl: what is the significance of the difference between
+mg and — m^? How can the quantization of both the
magnitude and orientation of orbital angular momentum be
explained in terms of fitting a wave to a spherical surface? For
more questions see 'angular momentum Questions.
Further information. See Chapter 2 of MOM for a discussion
of the quantum mechanics of a particle on a ring and on a
sphere: this is an introduction to the quantum theory of
orthogonal functions
161
BOX 14: Orbital angular momentum
Cfassicat definition I = r A p,
that is, i x = VP z ~*Py
l y =zp x xp z
%=*p y yp x
with magnitude i = (H 2 + S. 2 + S. 2 ) y '.
X y Z
Quantum definition
C = r A p={h/iJrA V,
that is.
K = M)1y£zfr
2„ = (li/i) fe
bx x bz*
In spherical polar coordinates these become
$ x = lh/i) (Sin0 ^ + cotf? cos0 A)
* 3 f\
£~ 2 =Fi 2 A 2
where A 2 is the legendrian (see Box 11).
Commutation relations
{%, e K l  fli^ fy, y = M x $ e , fij = ihB^
[£ 2 ,y =0 q = x,y,z.
Eigenvalues and eigenfunctions
I 2 ^J0, 0) = W + 1)h 2 ^ (9, 0) fi = 0, 1, 2 . . .
K ^Bm < fl « *> = mh ^Sm {B  *' m » K, 81 HZ.,
^Em^' ^' are t ' 1e ° s P nerica l harmonics: see Table 22.
Matrix elements. The only nonzero elements are:
«. m + 1 1$^ l8,ro) = {W2)J [8(8 + D m(m ±1)]
(fi, ro ± 1 lf£ l£, m) = + (ih/2)/ [8(8 + D ~ro(ro ± 1 )] .
Shift operators
*! = *,
±i8
y
raising operator:
8j8,m) = rW[8(8 + 1)
lowering operator:
m(ro + l)]lB,m
+ 1)
JHK,m> =
IW [£ffi + 1) 
 m{m 
1)]IC,m
1).
J
angular momentum. For a more general development, see
Chapter 6. A summary of the properties of angular momentum
is given in Box 14, and it should be observed that the wave
functions for systems of given orbital angular momentum are
the "spherical harmonics. In this connexion see Kauzmann
0957). For a detailed account of orbital angular momentum,
see Brink and Satchler (1968), Rose (1957), and Edmonds
(1957). Tinkham (1964) connects it with the rotational sym
metry of systems. For the role of orbital angular momentum in
chemical problems, explore atomic orbitals, hydrogen atom,
and atomic spectra and its ramifications.
orthogonal functions, Two functions ^, and \j/ 2 are
orthogonal if the integral _fdr»K^2 vanishes; therefore they
are orthogonal if their overlap integral is zero. A trivial
example would be the orthogonality of the Isorbital on two
widely separated hydrogen atoms. Another example is the
orthogonality of the 1s and 2s orbitals on the same atom, or the
orthogonality of a Isorbital on one atom to a 2pJTorbital on
the neighbouring atom in a diatomic molecule,
A general result of operator algebra is that eigenfunctions
of an hermitian operator are mutually orthogonal if they
correspond to different "eigenvalues. When an eigenvalue has
degenerate eigenfunctions these need not be mutually
orthogonal, but combinations that are orthogonal may be
formed by the Schmidt orthogonalization process. In group
theoretical terms, two functions are orthogonal if they belong
to different irreducible representations of the point group of
the system. Although orthogonality is a natural consequence
of the type of operators one encounters in quantum mechanics,
it is also a most desirable property because enormous numbers
of potentially difficult integrals disappear automatically.
162
oscillator strength
Questions. 1. What is meant by the term 'orthogonal function'?
What are some examples of orthogonal functions? When can
we be sure that a set of functions is mutually orthogonal? If
they are not orthogonal, what process can be used to recover
orthogonality? What grouptheoretical property guarantees
orthogonality?
2. Consider the functions exp \m<j) in the range < 4> < 2tt:
show that functions with different integral values of m are
orthogonal. Show explicitly that the "hydrogenatom 1s
orbital is orthogonal to the 2s and 2porbitals. Show that the
wavef unctions for a "particle in a onedimension square well
are mutually orthogonal. Consider the lowestenergy degenerate
wavefunctions of a "particle in a twodimensional square well
and show that it is possible to find either orthogonal or non
orthogonal linear combinations of the two degenerate
functions which continue to satisfy the Schrodinger equation
with the same eigenvalue. Sketch the form of some of the
combinations.
3. Orthogonalize the "Slater 2satomic orbital to the Isorbital
by the Schmidt procedure. This involves forming the sum
$u = 4>u + c\j/ tt and determining c so that the new 2sorbital
t/4s is orthogonal to \j/ Js , How may the Slater 3sorbital be
made orthogonal to </» u and ^i s ?
Further information. See MQM Chapter 4 for the proof that
nondegenerate eigenf unctions of hermttian operators are
orthogonal, and for other consequences of orthogonality, For
the grouptheoretical description of orthogonality see MQM
Chapter 5, Tinkham (1964), Bishop (1973}, and Wigner
(1959). For the Schmidt orthogonalization process see
McGlynn, Vanquickenborne, Kinoshita, and Carroll (1972).
oscillator strength. The oscillator strength f is a measure of
the strength of a transition and is the ratio of the actual
intensity to the intensity radiated by an electron oscillating
harmonically in three dimensions. Thus for such an ideal
electron the oscillator strength is unity, and for strongly
allowed transitions it is found that f lies in the neighbourhood
of unity. Oscillator strengths for several types of transition are
recorded in Table 8.
The oscillator strength can be calculated from two
directions, the theoretical and the experimental, and so its
importance lies in the connexion it provides between theory
and reality, as well as in its usefulness as a classification of the
strength of transitions.
1, The theoretical calculation of "electric dipoSe oscil
lator strengths is based on the expression f= Ae</nhvBle 2 ,
where B is the "Einstein coefficient of stimulated absorption.
For electric dipoie transitions 8 is equal to cf 2 /6e fi 2 . w ' tn d
the transit ion di pole; this implies that f= (2m /3e 2 h 2 )hvd 2 .
Therefore, if we can calculate the "transition dipoie dfor the
pertinent transition we can find f.
2. The experimental determination of the oscillator
strength is based on its relation to the "extinction coefficient
e{v) at the wave number V, through the formula / = 433 X 10"'
fdve{v). Therefore, if the extinction coefficient is known over
the range of wavenumbers the integral provides an experi
mental measure of ffor the transition.
An important theoretical rule predicts that the sum of the
oscillator strengths for all the electrons in an A/electron mole
cule is equal to N: this is the KuhnThomas sum rule. Thus
overall the hydrogen atom behaves like an ideal oscillator,
because the sum of the oscillator strengths for all transitions
away from the ground state is unity.
Questions. 1 . What does an oscillator strength measure? What
is the oscillator strength of a (threedimensional) harmonic
oscillator? What is the sum of oscillator strengths for all
possible transitions from the ground state of the hydrogen
atom? State the KuhnThomas sum rule for an A/electron
atom. How is the oscillator strength related to the extinction
coefficient for a transition? How is the oscillator strength
related to the transition dipoie moment?
2. Calculate the oscillator strength for the transition from the
rjth to the (n 4 1)th level of an electron in a onedimensional
square well, and for a onedimensional simple harmonic
oscillator. Compute the integrated intensity of the absorption
bands.
Further information. For the properties of the oscil lator
strength, its connexion with the extinction coefficient, and
the derivation of the KuhnThomas sum rule, see MQM
Chapter 10. See also Kauzmann (1957) for a discussion and
Eyring, Walter, and Kimball (1944). for applications in
overlap
163
moderate constructive overlop shorn] constructive overlap strong destructive overlap zero net overlop
FIG. 03. (a) Interference of waves (in black] leads to the resultant drawn in colour (since the interfering waves have different wavelengths we have
selected only a smafl domain), [bj Shows the analogous situation for overlapping s and patomic orbitals and (c) denotes the regions of overlap
pertinent to (b. Note that horizontal hatching implies constructive interference, and vertical hatching destructive.
photochemistry, see Wayne (1970) and Calvert and Pitts
(1966).
overlap. When two waves lap into the same region of space
they interfere and their superposition gives rise to a new wave
with an increased amplitude in regions of constructive inter
ference (where the two overlapping waves have amplitudes of
the same sign) and diminished amplitude in regions of destruc
tive interference (where the amplitudes have opposite signs
and so tend to cancel) (see Fig. 03a). Since the electron
distribution is determined by the square of the amplitude of
the wavef unction, the electron will be concentrated in
regions of constructive interference and will be to some extent
banished from the regions of destructive interference. Therefore
the overlapping of two wavefunctions can strongly modify the
distribution of the electron (or any other species).
When an sorbital is brought up to a porbital along the
latter's axis {Fig, 03b) the amount of interference increases
as the separation decreases, and if they are brought up with
the same sign of their amplitudes the constructive interference
increases and the electron accumulates strongly in the region
of overlap. As the orbitals move even closer together, the region
of overlap increases further, but the net amount of overlap
decreases because the sorbital begins to overlap the region
of negative amplitude of the porbital on the other side of
the node. When the two nuclei are superimposed the sorbital
overlaps the positive and negative lobes of the porbital
equally: on one side there is constructive interference and on
the other there is an equal amount of destructive interference.
As the sorbital continues on its passage through the nucleus
the overlap gradually becomes entirely destructive because the
amplitudes tend to cancel, and over a fairly complicated
surface do in fact cancel to give a node. As the sorbital moves
away the destructive interference disappears because where
one orbital is large the other is small, and eventually goes to
zero. Had the sorbita! been brought up along the line perpen
dicular to the axis of the porbital there would always have
been an equal amount of destructive and constructive overlap
at all distances.
A measure of the net amount of overlap is provided by the
overlap integral, which is zero when there is no net overlap
(either because there is no overlapping or because there is an
equal amount of constructive and destructive interference in
the regions of overlap) and is unity when there is perfect
164
overlap
overlapping (when the overlap of an orbital with itself is
considered). Orbitals with zero mutualoverlap integral are said
to be orthogonal. The overlap integral S is calculated by
taking the two functions at some point r, multiplying them
together to give i/£{r)i£ (r), and then integrating this product
overall space: S = Jarif*(r)^ b (r}. It should be clear that this
definition of the overlap integral conforms with the properties
we have described.
Questions. 1. What happens when two orbitals overlap?
Describe the change in the electron distribution that occurs
when two Isorbitals approach each other, pass through each
other, and then separate {let them approach with the same
sign of their amplitudes). Sketch the approximate value of the
overlap integral for this process as a function of separation.
What is the behaviour of the overlap as a Isorbital is brought
up to a Sd^orbital along {a) the yax\s and (b) the xaxis.
Sketch the form of the "hybrid orbitals that arise in each case
when the nuclei coincide.
2. The overlap of two hydrogen Isorbitals separated by a
distance R is given by the expression
S(1s, Is) = (1 + R/a +RV3al) exp{R/a ).
a is the Bohr radius (53 pm, 053 A). Plot this function as a
function of R, At what separation is S a maximum? The over
lap integral of a Isorbital and a 2sorbital or a 2porbita!
approaching along the latter's axis is given by
S{1 s, 2s) = {1/2/ 3){1 +R/a + 4flV3ag + R 3 /3al) exp(ff/a )
Sits, 2po) = {/?/2a )(1 + R/a +R 2 /3a 2 ) exp{R/a ).
Plot these functions, and find the position of maximum overlap
in each case.
3, Suppose that on one of the nuclei we have a "hybrid orbital
of the form i^2 S (r) sin£+ ^2 P (r)cos, where sin £ and cos £ are
mixing coefficients, and we bring up a Isorbital along the
axis of this hybrid. Discuss the form of the overlap integral;
sketch its dependence on R for a mixture of 2s and 2p that
gives the maximum overlap at all separations. Plot this
optimum £ as a function of R .
Further information. Overlap is of importance in all discussions
of bonding; therefore see MQM Chapter 9, and especially
Coulson's The shape and structure of molecules {OCS 9),
Coulson 0961), and (vlurrell. Kettle, and Tedder (1965). The
manner of calculating overlap integrals is described in Eyring,
Walter, and Kimball (1944) and in more detail in McGlynn,
Vanquickenborne, Kinoshita, and Carroll (1972). Extensive
tables of overlap integrals have been published in a number of
places: see the list of references on p. 421 of the book by
McGlynn etal (1972). Overlap is considerably enhanced when
the orbitals involved are 'hybridized, and the strength of the
bond so formed is increased. The occurrence of overlap, and
the consequent interference effects, are aspects of the "super
position principle, which is one of the fundamental aspects of
quantum mechanics.
p
pairing. Electrons pair when they enter an orbital with
opposed spins. Electrons must pair if they are to enter the
same orbital, for that is a requirement of the "Pauli principle.
Electron pairs are of prime importance in the theory of the
chemical bond, and at an elementary level it is sometimes said
that bond formation reflects the tendency of electrons to pair.
That shorthand must be based on the deeper remark that
electrons will sink to lower energy if they do pair, and the
reason for this may be found by considering the formation of
a bond. The first electron of a bond enters the region of
lowest potential energy (between the two nuclei), and the
second can enter the same region only if it has opposed spin
and pairs with the first; because of the Pauli principle a third
electron cannot enter the same region, and so it will be much
less effective in bonding. The basis of this description may be
illustrated by molecularorbital theory, for in order to be
most effective in bonding an electron must enter a bonding
orbital (which is a distribution concentrated between the two
bound nuclei); for two electrons to enter their spins must be
opposed; three or more electrons cannot enter this best,
bonding orbital and are forced to enter higherenergy orbitals,
which may be antibonding. Hence the great importance of
electron pairs is merely a manifestation of the fact that only
by pairing are two electrons able to enter the lowestenergy
orbitals and be most effective in bonding, and more than two
cannot get into the most favourable orbital; and the tendency
of electrons to pair is a reflection of their tendency to seek
lowestenergy situations,
A slightly different situation holds in atoms, for there the
Hund rules tell us that the outermost electrons tend not to
pair: but here there is only one attractive centre, not two or
more, and the important effect is the operation of exchange
and "spin correlation.
Further information. See bond and molecular orbital for a
guide to the role of pairing, and the Pauli principle for its
basic source. The valencebond theory is an approach to
chemicalbonding theory that recognizes the importance of
pairing at the outset, and develops that point of view
quantitatively; refer to its entry for details.
particle in a square well. A particle constrained to remain
strictly within a particular region of space, with no seeping
into or through the walls of the container, is a particle in a
square well. It is so called because the confinement can be
achieved by arranging a potential to be zero throughout the
domain of freedom of the particle but to rise perpendicularly
to infinity at the edges. The geometrical shape of the domain
may take a variety of forms: a simple example is the one
dimensional square well, or box, where the particle can travel
freely from to £ along the xaxis, but be nowhere else. The
twodimensional square well may take any shape in a plane,
and the rectangle or the square are particular examples. The
threedimensional square wells include the cube and the sphere
(which is sometimes called, without intention of paradox, a
'spherical square well').
In each case the energy of the particle arises entirely from
its kinetic energy, because it cannot penetrate the region
where the potential energy differs from zero. Since the
particle is confined its energy is quantized, and the permitted
165
166
particle in a square well
BOX 15: Particle in a square well
Linear (onedimensional) box of length L :
n = 1,2
Rectangular, threedimensional box of sides t j, L^, Ly.
/>i= 1, 2, . . .; n 2 = 1, 2, . . .; n 3 = \,2... ..
Spherical box of radius R
^ f U\2mR 2 )
f l0 = 3142, f n = 4493, f a = 5763, f 20 = 6283,
f l3 = 6988, f tl m 7725, . . .
Ar as f/R;/^ is a spherical Bessel function. See §36 of
Davydov (1965).
energies (which are set out in Box 1 5 for various box shapes)
are obtained from the "Schrddinger equation for the problem
and the boundary conditions characteristic of the well. From
the requirements that the functions be continuous and the
probability of finding the particle anywhere outside its domain
of freedom be zero, one deduces that the wavefunction must
always vanish at the walls. From this it follows that although
the permitted functions are the same as freeparticle functions
within the box, their wavelength must satisfy /?A/2 = L, with
n an integer greater than zero; otherwise their "nodes would
not occur at both walls. Since the wavelength determines the
momentum by the c de Broglie relation, and since the
momentum determines the kinetic energy, it follows that the
permitted energy levels are confined to the values given by
n IL . This situation is a good example of the discussion of the
quantumtheoretical significance of "kinetic energy and the
curvature of the wavefunction for, as we attempt to cram more
waves into a given length, the function has to be more buckled,
and its curvature greater. The wavefunctions themselves are
just ordinary sine waves of decreasing wavelength, and a few
of them are illustrated in Fig. PI.
FIG. PI . Energy levels and wavef unctions for a particle in a one
dimensional square well.
We may summarize the properties of a particle in a square
well in the following way.
1. In a onedimensional box the energies are determined by
a single quantum number n and are proportional to n 2 /L 2 (see
Box 15)./? may take the values n = 1, 2, 3, . . ., and since
n = is forbidden (the wavefunction with n = would vanish
everywhere) the lowest energy is greater than zero; therefore
there is an irremovable zeropoint energy. All the states are
nondegenerate.
2. In a threedimensional box the energies depend on
three quantum numbers, and in a rectangular box of sides
Pa sc henBack effect
Li,L 2 , i. 3 are proportional to (n\ IL\) + (nllL\) + (nllil),
with n ,, n 2 , n 3 allowed all integral values above zero. There
is a zeropoint energy corresponding ton l = n 2 = n i = 1.
3. The wavefunctions are simple sine functions of
different wavelengths {and therefore of different kinetic
energies). The zeropoint energy corresponds to a state in
which the particle is most probably near the centre of the
box, and furthest from the walls. At higher quantum numbers
the probability density is spread more evenly throughout the
region.
4. The energy levels move apart as the walls become more
confining: an integral number of half wavelengths must be
fitted into a decreasing length; therefore the wavelength
must decrease, and so the kinetic energy increases. As the
confining walls move apart the energy separation diminishes,
and in the limit of infinite separation the levels form a
continuum, and the system is unquantized.
5. Increasing mass has an effect similar to the effect of
increasing the size of the domain: a particle of large mass
behaves more classically than one of low mass for a box of
given size.
6. A localized particle may be constructed by forming a
suitable "superposition corresponding to a "wave packet at
the point where the particle is located; the packet so formed
moves in close accord with the predictions of classical
mechanics for a particle confined to a region surrounded by
infinitely rigid and perfectly reflecting walls.
Questions, 1 . What is the meaning of the expression 'square
well? What is the effect of the presence of impermeable
walls on the allowed energy levels of the system? Why is the
energy proportional ton 2 //. 2 ? From the classical expression
for the kinetic energy in terms of the momentum, and the
•de Broglie relation for the momentum in terms of the
wavelength, deduce an expression for the energy of a particle
in a onedimensional square welt, and compare your answer
with Box 15, What is the lowest energy permitted to a
particle in a box, and what is its dependence on the size of
the box? What interpretation in terms of the "uncertainty
principle may be put on this zeropoint energy? What is the
mean momentum in any single energy state of the box? In
what limit does the particle behave in a classical manner?
167
Calculate the lowest energy of an electron in a box of length
1 m, 1 nm, CM nm, 1(T ls m: what is the energy of transition
to the first excited level in each case? Why is a fly in a room a
classical particle, to all intents and purposes (if flies have
intents or purposes)? An electron in a conjugated polyene
may be considered to be a particle in a onedimensional box:
estimate the transition energy from the/jth to the {n + 1)th
level in a chain of N carbon atoms,
2. The "Schrodinger equation for a onedimensional box is
(1rV2m)(d 2 /dx 2 )0= E\p; solve this equation in terms of a
function of the form Asm kx + Scos kx, and apply the
boundary conditions that the function vanishes at x = and
x = L, the edges of the box. Show that A may be deduced
from the fact that the functions should be 'normalized, and
show that states of different energy are "orthogonal. Show
that a rectangular twodimensional box may be solved in terms
of 2 ondimensional boxes by the method of separation of
variables (see "Schrodinger equation).
Further information. See MOM Chapter 3 for detailed infor
mation about the solution of the Schrodinger equation for
particles confined in boxes. The spherical square well is more
difficult, but its solution is outlined in Kauzmann (1957), and,
since it is a fair model of an electron in liquid ammonia (metal
ammonia solutions), see LePoutre and Sienko (1964) and
Lagowski and Sienko (1970) for its applications and properties.
When the barrier is not infinite the particle can seep into the
walls, and if the potential falls outside to some finite value the
particle might 'tunnel through the barrier: situations of this
kind are discussed and solved in §32 of Davydov (1965),
Schiff (1968), Messiah (1961), and Landau and Lifshitz
(1958 a). Gol'dman and Kryvchenkov (1961) work through a
number of problems involving barriers. See also Chapter 1 1 of
Bohm (1951 ) for a good discussion of square potentials.
PaschenBack effect. The PaschenBack effect is the de
coupling of spin and orbital angular momenta by an applied
magnetic field.
Consider an atom with both "spin and orbital angular
momenta, s and I; these are coupled together by the "spin
orbit coupling interaction and form a resultant angular
momentum which is represented by the "vector j. Both s
168
FIG. P2. PaschenBack eCfed.
and I precess around their resultant, and in a magnetic
field of moderate strength the resultant vector also
precesses about the field direction (Fig. P2 a): the energy
of this interaction gives rise to the ""Zeeman effect. As the
strength of the applied field is increased the strength of its
interaction with the spin and orbital "magnetic moments
becomes so great that it begins to overcome the spinorbit
coupling. At this point the spin and orbital moments begin
individually to precess around the direction of the field,
and the spinorbit coupling is broken; the motion is complex
because the spinorbit coupling and the applied field are in
competition for the two moments. If the field is made
sufficiently strong (of the order of tens of kilogauss) the
battle is resolved in its favour, and the state is one of almost
pure precession of each type of momentum about the
field's direction (Fig. P2 b). The PaschenBack effect has
succeeded in decoupling the momenta.
The spectral consequence of the effect is that the elec
tronic transitions occur in a simple fashion: the optical
field interacts with the orbital angular momentum and
causes transitions which are independent of the spin
direction. The anomalous 'Zeeman effect, which depends
on the interplay of spin and orbital moment effects, is
replaced by the normal Zeeman effect, characteristic of
systems without spin.
The term PaschenBack effect is also used for other
situations in which the spin and orbital momenta are
decoupled by a field; one example is the effect of the
axial nuclear Coulombtc electrostatic potential on the
momenta in a diatomic molecule; its effect is to modify the
"Hund coupling case.
Questions. What is the PaschenBack effect? Discuss the
effect in terms of the "vector model of the atom. Why does
the effect simplify the anomalous Zeeman effect and
replace it by the normal effect? If the spinorbit coupling
energy in an atom is of the order of 100 cm" 1 estimate the
strength of the applied field required to decouple the
angular momenta.
Further information. See the discussion of the 'Zeeman
effect and MQM Chapter 8. See also §11.3 of Herzberg
(1944) and §111 F3 of Kuhn (1962) for detailed accounts
of its spectroscopic consequences.
Paul! principle. The Pauli exclusion principle states that no
more than one electron may occupy a particular state: a con
sequence of this is the "aufbau principle which underlies the
periodicity of the elements; for an atomic orbital labelled
n, 6, mn, may be populated by no more than two electrons and
these must differ in the quantum number m (and so must have
opposite spin orientations).
The exclusion principle is a special case of the full Pauli
principle which makes a general statement about all particles.
The Paul! principle starts from the view that if two particles
are indistinguishable, when they are interchanged the calcu
lated properties of the system must remain unchanged. In
particular, since the particle density is proportional to i^l 2
(see wavef unction), when two indistinguishable particles are
interchanged (the first put where the second was and vice
versa) the particle density, and hence l^l 2 , must not change.
This implies that \jj itself can either change sign when the
particles are interchanged, or not change sign; no more
complicated change is permissible. The change of sign, or lack
of it, must occur for all possible pairs of indistinguishable
particles in the system when the two particles of each pair are
interchanged. A fundamental distinction which determines
whether or not the wavefunction changes sign is between
•fermions (which are particles with halfintegral spin, such as
electrons, protons, neutrons, 13 C nuclei, and 3 He) and "bosons
(which have integral spin, such as 2 H, 4 He, 12 C, and photons).
The principle states that if a collection of identical particles
1,2 n is described by the wavef unction ip{r h r 2 r )
then the form of this function must satisfy a stringent
requirement: If the particles are fermions the function must
change sign whenever the positions of any two particles are
interchanged; whereas if the particles are bosons the function
must not change sign. Another way of saying this is that under
particle interchange the total wavef unction for identical
fermions must be antisymmetric, and for identical bosons it
must be symmetric.
The principle is adequately illustrated by considering a two
particle state Mr,, t 2 ), or ^(1, 2} for short: if the particles are
fermions Nature demands that i£{2, 1) = ^1, 2). The impli
cation of this for electrons (fermions) is as follows. Suppose
we have a state \j/ a which can be occupied by electron 1 and a
state ^ which can be occupied by electron 2, then the total
system is described by a function of the form $ (rj^ (r 2 ), or
^,(1)^(2) for brevity. But this function does not satisfy the
Pauli principle because it is neither symmetric nor anti
symmetric in the labels 1, 2 (it is unsymmetric). We can turn
it into an acceptable function for electrons by replacing
^,{1)^(2} by \pJ1)\p b i2)  lA a (2)^ b (T), for when 1 and 2
are interchanged the sign of the function changes. This
function demonstrates why only one electron can exist in a
given state: if the states ^ a and ^ b were identical, so that
both electron 1 and electron 2 were in the same state, the
form of the function that satisfies the Pauli principle would
be ^0)^(2} ^(2)^(1), which vanishes: this shows that
multiplyoccupied states vanish and that the exclusion
principle is a special case of the full principle. If instead of
electrons we were dealing with bosons, the appropriate
two particle state would be ^ a (1)^ b (2) + ^ (2)^ (1), and
this does not vanish if $> = $ < therefore bosons do not
satisfy the exclusion principle because multiplyoccupied
states are permitted (a large number of photons may occupy
the same state and give rise to an intense monochromatic
light beam). Remember that the Pauli principle applies to the
total state of the system, so that ij/ g and & represent space
(orbital) and spin states.
The Paul! principle was introduced in order to account for
the spectrum of helium, for an analysis of its spectrum showed
penetration and shielding
169
that many expected "terms were absent, and their absence
could be explained on the basis that two electrons on a given
atom could not occupy the same state. The principle can be
given a theoretical foundation, and Pauli, by considering the
problem relativistically, and demanding that the energy of a
system be positive, showed that all particles of ha If integral
spin must have antisymmetric wavefunctions and behave as
we have described. Possible exceptions to the principle are
quarks, which are peculiar particles used in one theory of
elementary particles; but as they have not yet been observed
experimentally they may be figments of the imagination, and
particles, not figments, are required to satisfy the Pauli
principle.
Without the Pauli principle matter would not be rigid;
likewise, if electrons had no spin (and were bosons) matter
would have less bulk, everything would be denser, nothing
rigid, and everything very sticky.
Further information. The experimental basis of the Pauli
principle is described in MQM Chapter 8. A product of
functions may be made to accord with the principle by
writing it as a "Slater determinant, and the discussion is taken
further in that entry and in the one on "antisymmetric
functions. Problems too will be found there and in Further
information. Pauli deducing his principle may be observed in
Pauli (1940); for quarks see "fermions.
penetration and shielding. Like Castor and Pollux,
penetration and shielding seem inseparable twins: so they are
in application but not in contemplation. Let us fix our ideas
by considering the sodium ion Na + , which consists of a
strongly charged nucleus (Z= 11) surrounded by ten electrons
filling the K and L shells (the configuration is 1s 2 2s 2 2p 6 }.
Drop an electron into the 3s "orbital and observe that, since
the sorbitals all have a nonvanishing probability of being at
the nucleus, the electron penetrates the surrounding electrons
to a small extent, and with a small but nonzero probability
may be found in the vicinity of the nucleus. In that region it
will have a low potential energy and be stabilized. If instead of
dropping it into a 3sorbital we contrive to deposit it in a
3porbital, the electron is unable to penetrate so closely to the
nucleus {porbitals have modes at their nucleus), and so it does
170
perturbation
Election
density
Distance from nucleus
FIG. P3, The coloured lines show the 3s and 3p radialdistribution
functions superimposed on K, L shell electron density.
not attain the region of lowest potential energy (Fig. P3). That,
then, is penetration; we can believe that sorbitals might lie
lower in energy than p, dorbitals on account of their closer
approach to the nucleus.
Close approach is not of itself sufficient to lower the energy,
because in the hydrogen atom all the orbitals with the same
principal quantum number carry the same energy, irrespective
of whether they are s, p, or d. In order for penetration to have
an effect it is necessary to eliminate the peculiar property of
the pure Coulomb interaction which is responsible for the
unique properties of the hydrogen atom. This is achieved by
the shielding effect of the other electrons that are present in
the case of manyelectron atoms (see Fig. P3). In these the
potential experienced by our extra electron is that of a nucleus
of charge Ze(Z= 1 1) at small radii, but at larger radii, when
the test electron is outside the core, the potential is more
characteristic of a nucleus of charge e, because the (Z — 1 )
electrons have shielded all but one unit of positive charge.
Therefore the potential of the shielded nucleus drops off far
more rapidly than the Coulombic law would entail, and so the
peculiar characteristics of the Coulombic potential are
eliminated. Consequently the s, p„ and delectrons possess
different energies.
It should be clear that the different orbitals have different
energies on account of the shielding effects, and the order of
their energies and their separation depends on the extent of
their penetration of the core.
Questions. Which orbitals penetrate most closely to the
nucleus? What is the essential role of shielding? What
accounts for the fact that selectrons generally lie lower in
energy than pelectrons, and pelectrons lie lower than
delectrons? What is the order of 'ionization potentials for
s, p, and delectrons?
Further information. See MQM Chapter 8 for a description of
the extent and role of penetration and shielding. Simple
diagrams are given there, and in White (1934) and Herzberg
(1944), of the extent of penetration of various Inner shells by
various outer electrons. For a discussion of the screening
constant see "Slater atomic orbitals. The effects of pen
etration and shielding are most important for the structure of
the periodic table, for they influence the properties of the
elements through the 'aufbau principle and the ionization
potential. These matters are discussed by Puddephatt in
77je periodic table of tfie elements (OCS 3) and by Phillips
and Williams (1965).
perturbation. Most systems of interest are described by
Schrodinger equations too difficult to solve exactly; but
inspection of the problem often reveals that a simpler,
solvable system closely resembles the true, intractable system.
If the "wavef unctions and energies of this simpler system are
known it is possible to adjust them so that they are distorted
in the direction of the true wavefunctions and energies. If the
true system resembles the simpler system very closely the
amount of distortion required is very small, and is a mere
perturbation of their form. The modification of the simple
wavefunction can be achieved by mixing into it other wave
functions of the simple system in the appropriate proportions,
and perturbation theory provides the recipe for the mixture.
At the same time, perturbation theory shows how to cal
culate the additional terms that must be added to the energy
of the simple system to yield the energy of the true system.
Firstorder perturbation theory teaches that the wave
function of the simple system \p should be modified to
&o + C& + c 2 4/ 2 +   n w here tne $„ are tne v a rious waue "
functions of the model system corresponding to the energies
£,, and where the coefficients^ are determined by the
ratio of the strength of the perturbation (which is essentially
the energy difference between the true and the simple systems,
and is actually the 'matrix element of the perturbation
hamiltonianthe difference between the true hamiltonian
and the simple hamiltonian) to the energy separation E — E
The wavefunction obtained in this way is the firstorder
wavefunction.
An example to fix our ideas at this stage could be the
distortion of the ground state of the hydrogen atom by an
applied electric field (the "Stark effect): the simple system
would be the hydrogen atom in the absence of applied fields
(the wavefunctions and energies, of course, are known), and
the distortion ("polarization) of the atom by the applied field
could be taken into account by adding some 2p orbital into
the ground state, and then some 3p^ orbital, and so on. The
coefficient of each orbital is determined by the ratio of the
perturbation energy (in this case the electrical interaction
energy ezE, which comes from the expression d.E for a
dipole d in our electric field E) to the energy separation for
each orbital (£ ls £ 2p , E ls f , p , . . .) , The proportion of
the orbital in the mixture is determined by the square of the
mixing coefficient (see "superposition principle), and so it is
clear that only those orbitals lying fairly close in energy to the
1sorbital will be mixed significantly. This is a general result of
perturbation theory: the greater the energy separation, the fess
the mixing. It follows that if all the energy levels lie far above
the ground state, the simple system cannot be distorted very
much, and so it will resemble very closely the true system.
Conversely, if the perturbation is too strong (the true and
simple systems being very dissimilar) a large amount of mixing
may occur and the firstorder theory might be inappropriate:
the solution is generally to choose a better model system.
It should be noticed that the form of the true wavefunctions
emerges automatically from the perturbationtheory
machinery, for the recipe generates the correct distortion. This
is in contrast to the "variation approach to the approximation
of wavefunctions, where the final form depends on the
original guess of the form of a sufficiently flexible function.
The modification of the energy due to the perturbation
may also be calculated, and although the firstorder energy is
sometimes sufficient, it is normally necessary to calculate the
perturbation 171
secondorder energy. The former is calculated by taking the
perturbation energy and calculating its average value over the
undistortsd wavefunctions of the simple system. The second
order energy correction takes into account the distortion of
the simple wavefunctions by the perturbation: first the wave
function is distorted by the application of the perturbation,
and then the average value of the perturbation energy is
calculated over this distorted wavefunction. The name
'secondorder' indicates that the perturbation is involved
twice,
A helpful analogy, which enables one to appreciate how
perturbations operate, is the mutilation of a violin string by
suspending from it a number of small weights. The weights
hanging from the nodes affect neither its motion nor its
energy, but those hanging from the antinodes (the points of
maximum displacement) may have a profound effect on its
vibrational energy and waveform. The firstorder energy
correction is found by averaging the effect of the weights (the
perturbation) over the wavefunction (in this case the dis
placement) of the system. The weights also distort the wave
form of the string: the nodes are slightly shifted and the pure
sinusoidal shape lost. This distorted mode can be reproduced
by taking a suitable superposition of the harmonics of the
unladen string, and if the weights are not ton big it should be
clear that only a few of the harmonics need be incorporated
to give a good representation of the distortion. The second
order energy correction to the vibrational energy of the string
can be found by averaging the effect of the weights over the
distorted waveform. This secondorder correction should be
added to the firstorder correction, and their sum added to the
original vibrational energy of the unladen string, in order to
find a good approximation to the true energy of the
modified string.
The order of magnitude of the correction to the energy
brought about by a perturbation of energy P is ~/> for the
firstorder correction, and ~P 2 /A for the secondorder
correction, A being a typical energy separation of the undis
torted system. Often the firstorder correction disappears
identically (on grounds of symmetry). The firstorder
correction to the wavefunction yields coefficients of order
P/A, and so the proportion of states admixed is of the order
P /A 2 . The application of perturbation theory is normally
172
perturbation
valid so long as P is much smaller than A: that is, so long as
\P/A\% 1. It is important to note that the secondorder energy
can be obtained from the firstorder wavef unction (in general
the nthorder energy can be obtained with knowledge of the
(n — 2)thorder functiont); thus jubilation at obtaining a good
energy should be tempered with the reflection that the wave
function itself might still be very poor, Alternatively, the more
sanguine will reflect with some satisfaction that such a good
energy can be obtained from such a poor wavef unction. (For
example, a wavefunction correct to within 10 per cent can give
an energy correct to within' 1 per cent.)
The algebraic recipes for perturbation theory are set out in
Box 16.
Timedependent perturbation theory allows the perturbation
to vary with time and permits the calculation, as an important
application, of the effect of the perturbation caused by a light
wave. The distortions change with time and the admixture of
the excited states of the system may be interpreted as tran
sitions from the ground state to another state. See transition
probabilities.
Questions, 1. What is a 'perturbation'? What does it affect?
How may the effect of the perturbation of the wavefunction
be taken into account? What is the order of magnitude of the
coefficients of the admixed functions, and in what proportion
are they mixed? How is the firstorder correction to the energy
calculated? How is the secondorder correction calculated?
Why do they differ? What action should you take if you find
that the wavefunction of the simple system has to be severely
modified? What is the order of magnitude of the first and
secondorder energy corrections and the firstorder wave
function distortion when a perturbation of energy equivalent
to 10 J mot' 1 is applied to a system in which the energy
separations are of the order of 1 kJ mof l ? A mass of 1 kg
hangs from a spring of forceconstant 1 N m" 1 : how will its
motion be modified by the addition of a 100 g mass
ls=98m s~ 2 \? Now consider a similar quantummechanical
problem: let a 2,0 Po atom be oscillating against an effective
"forceconstant of 40 N m" 1 , and let it be in the vibrational
t Really, if the lower orders don't set us a good example, what on earth
is the use of them? (Oscar Wilde. The importance of being earnest,
Actl.)
BOX 16: Perturbation theory
We suppose that we know the eigenvalues and eigenfunctions
of the hamiltonianW (0) :
wloi^W = £(°)iM >
r n n r r» '
and we require the energies and wavefunctions of a
hamiltonian W:
where W< " is a firstorder correction (of order X in some
small parameter X) and rV' 2 ' is a secondorder correction
(of order X 2 ). The true energies and wavefunctions are
written
f = £<0 l + f (l) +f {2) + _
m m ttt m
\b =li/< » 0< l >+ lir< 2 >+.:„
*vn r m 'm r m
Zerothorder energy and wavefunction
E m =<^\H l0) l^ 0) > corresponding to ^ 0) .
Firstorder correction
f i !) = <C lwl,, C ,)
m
0**> = T,' e ^"l, where c = J ll*St
[ m n )
Secondorder correction
+ *\ <c4°>> ffl r
Time dependen t perturba tion theory
We suppose that H = H w + H ll Ht)
and write
* m W = E c „ W 1 W* *1 0) W = tfT exp (i£<°>f/r.).
(if *^m^ it n n ft ft
n
Then
0„{t) = e n i0)(mZfl dtWW&W exp fa^ .
phonons
173
where
and
w^ , ( f ')=<c w(i)f ' )i ^r )
nk n k
For weak perturbations applied to a state in which
initially c.(0) = 1, c f {Q) * (only state / occupied), then to
first order:
c ; (t) = 1
e,w = (r/h>£df'wk»(r')wcp i^r'.
"1 — ex p \\{iti fl + to)t] 1 — exp [ i (uy f; — cj) f f
c f M=V fi
ll(CJ ff + Oj)
H<*> fj ~ CO)
and if ui f .  oj < GJ ff + to the probability of being in state
fis
4V V
sin 2 (oj ff  6j}f,
ground state. Let it emit an aparticle. What is the order of
magnitude of the probability of finding the resulting ^Pb
atom oscillating in its ground state?
2. The quantummechanical expressions for the perturbation
corrections to the energy and wavef unction are given in
Box 16. Take a "particle in a onedimensional squarewell
wavefunction and add to the system a perturbation of the
form —qx, <x < L: find the firstorder energies and wave
functions. Using the same model, select a system that has a
flat potential within the walls except for a small rectangular
dip of depth D and width W. Set the centre of this dip at the
centre of the well, and let W< L. Calculate the firstorder
energy correction to a particle in the ground state [n = 1 ),
and then in the first excited state {n = 2). What do you notice
about the extent of correction in each case? Is the difference
also reflected in the correction to the wavefunction? Now
slide the centre of the dip to* = LIZ; what happens?
3. Estimate the extent of distortion and the correction to the
energy that results when an electric field of 10 6 V m~' is
applied to a ground state "hydrogen atom.
Further information. See MQM Chapter 7 for the details of
perturbation theory. Perturbation theory is one of the most
important methods for calculating atomic and molecular
properties: therefore see MQM Chapter 11, Eyring, Walter,
and Kimball (1944), Hameka (1965), Davies (1967), and
Kauzmann (1957). The mathematics of perturbation theory
and its recent developments are described by Hirschfelder,
ByersBrown, and Epstein (1964) and by Wilcox (1966). The
last two references describe the differences between the
RayleighSchrodinger perturbation theory (which is the scheme
set out in the Box), and the WignerBriiiouin perturbation
theory. The convergence of a perturbation expansion to the
exact energy is a frisky problem, slightly tamed by the
RellichKato theorem described on p. 6 of Wilcox (1966).
Timedependent perturbation theory enables, among other
things, the evolution of a wavefunction to be calculated as a
function of time; see transition probability, Chapter 7 of
MQM, Davydov (1965), and Heitler (1954). An account of
Heftier 's theory has been given by Hameka (1965). A recent
review of timedependent perturbation theory is that of
Langhoff, Epstein, and Karplus (1972).
phonons. Just as photons are "quantized vibrations of the
electromagnetic field so phonons are quantized vibrations of a
crystalline lattice. Imagine first a linear chain of atoms which is
vibrating in a lowfrequency mode (Fig. P4): this vibration is
quantized and may possess only discrete amounts of energy;
if its frequency is V its energy must be some integral multiple
of hv. Instead of exciting a single vibrational mode through its
successive evenlyspaced quantum levels it is possible to regard
the rising energy of the system as resulting from the addition
of hypothetical particles to that state; if n particles enter a
state of characteristic frequency V the energy of the system
rises by nhv. This particle picture of excitation of vibrations
is the basis of the concept of a phonon: a phonon is a
quantum of excitation of a specified frequency, and may be
envisaged as a particle of energy At 1 added to the system.
In a complex solid, phonons of different frequency exist
(just as in the electromagnetic field light of different fre
quencies exists). As in the case of light the phonons in a solid
may be polarized, but as well as transverse phonons, in which
the lattice atoms are displaced perpendicular to the propagation
direction (Fig. P4 a), longitudinal phonons, in which the dis
placement is along the direction of propagation, as in a sound
174 phosphorescence
framvene mode
longitudinal mode
wave in a fluid, rnav also exist (Fig. P4 b). A further compli
cation may be illustrated by the case of an tonic lattice con
taining positive and negative ions. First imagine the vibrations
of the two interpenetrating lattices as independent: now con
sider the combined system. Two situations may arise (see
Fig, P5). In the first the lattice of positive tons moves in phase
with the lattice of negative tons: this gives rise to the
acoustical branch of the phonon spectrum. In the second the
phase of the displacement of the two lattices is opposite: this
gives rise to the optica/ branch. The latter name arises from the
modulation of the lattice dipole moment during the outof
phase movements of the two sublattices, and the fact that a
light beam can interact with this oscillating dipole and so
stimulate that particular branch of the lattice vibrations.
ccowric branch
Q ©
n
©r © e ©
© ©
€X>€MD
®
© ©
optical branch
®
®
®
©
FIG. P5. Acoustic and optica) branches of the same wavelength. Note
that the dipote changes only in the latter.
3+i
2+i
l+l
0+t
Energy/hy
3 phonon 5
2 phonom
I phonon
no phonons preienr
FIG. P4. Transversa and longi
tudinal modes □( the same wave
length: the extent of excitation
can be expressed as the number
of phonons present.
Further information, A good introduction to lattice
vibrations is given in Kittel (1971) and a more advanced one
by Kittel (1963) and Ziman (1960, 1972). The role of
phonons in determining the electrical, optical, and thermal
properties of solids is discussed in these books. For a simple
account of the matter see Jennings and Morris's Atoms in
contact (OPS 5).
phosphorescence. When a phosphorescent material is
illuminated it emits light, and the emission may persist for an
appreciable time even after the stimulating illumination has
been removed; in this sense phosphorescence differs from
■fluorescence, for in the latter the emission ceases virtually
instantaneously. There is also a mechanistic distinction
between the two processes, and this is connected with the
persistence of the emission.
Phosphorescence occurs by the following mechanism: the
groundstate molecule, in which all the spins are paired (and
which is therefore a 'singlet state Sn), absorbs the incident
light and makes a transition to an upper singlet level. The
electronic excitation is accompanied by a vibrational
excitation (see Fig. P6), and this vibrational energy is trans
ferred to the surrounding molecules by the type of processes
described in "fluorescence. Indeed, the molecule is well along
the path that leads to fluorescence; but if the vibrational
deactivation is not too fast another process may intervene.
Let there be a triplet state of the excited molecule (in which
two spins are unpaired): this is illustrated as the curve T, in
the diagram. There is a nonvanishing probability that the
molecule will switch from the singlet state to the triplet as
phosphorescence
175
FIG. P6, The mechanism of phosphorescence,
it steps down the vibrational energy ladder (this is the
intersystem crossing, ISC). At the foot of the ladder it finds
itself trapped. It cannot radiate its electronic energy and drop
to the ground state because that involves a singlettriplet
transition (which is forbidden by the "selection rules). It
cannot clamber back to the crossing point and then step down
the singlet ladder, because the collisions with the lattice cannot
supply enough energy. It cannot give up its electronic energy
to the surrounding molecules by a radiationless transition,
because we have supposed that even the vibrational deacti
vation is weak, and that involved a smaller energy {which is
easier to remove). If the foregoing description were true, the
molecule would be stuck in the upper triplet. But the remarks
are not strictly true, like most remarks, and the important
fallacy is that the singlettriplet transition is forbidden. If it
were strictly forbidden the molecule would have been unable
to cross from the singlet state to the triplet. The fact that it
did cross implies that there is enough "spin orbit coupling
present to break down the singlettriplet selection rule, and so
this becomes weakly allowed. But as it is only weakly allowed
the transition T— *S is slow, and may persist even after the
illumination has ceased.
Phosphorescence involves a change of "multiplicity {an
unpairing of spins) at an intermediate step, and this is its
mechanistic difference from "fluorescence. From this point of
view it follows that phosphorescence may occur if there is a
suitable triplet state in the vicinity of the excited singlet states
of the molecule and if there is a sufficiently strong spinorbit
coupling to induce ISC: a heavy atom enhances this crossing
probability {the heavyatom effect ) . Furthermore, there must
be enough time for the molecule to cross from one curve to
the other, and this means that the vibrational deactivation
must not proceed so fast that the molecule is quenched and
taken below the point where the curves intersect before the
ISC interaction has time to operate. It is for this reason that
many molecules which fluoresce in fluid solution are found to
phosphoresce when they are trapped in a solid lattice, such as
a gel or glass. We can also predict from Fig. P6 that the wave
length of the emitted light should be longer (further into the
red) than fluorescent emission: the lowest vibrational level of
the triplet lies below that of the lowest excited singlet. Finally,
it is not impossible for some molecules to clamber back into
the singlet Si, and to fluoresce into the ground singlet: this is
stow fluorescence, the triplet state acting merely as a reservoir.
What is the evidence that the triplet state is involved in
phosphorescence? The first direct evidence came from the
determination of the "magnetic susceptibility of a phosphor
escent sample with and without illumination: it became
paramagnetic when the light was on. The most sensitive
procedure is to apply "electron spin resonance to the phos
phorescent state: this shows unequivocally that a triplet
state is involved.
Questions. How does phosphorescence differ observational ly
from fluorescence? Sketch the diagram corresponding to the
formation of a fluorescent state and superimpose it on the
diagram that leads to phosphorescence. What competitions
determine which path is taken? How may phosphorescence be
enhanced relative to the fluorescent and radiationless decay
routes? What is meant by the term 'inter system crossing', and
how does it differ from 'internal conversion'? Why is phos
phorescence a slow process? What perturbation is responsible
for the ISC and the emission? How may this perturbation be
enhanced? What is slow fluorescence, and how may it be
176
photoelectric effect
distinguished from phosphorescence? How may it be dis
tinguished theoretically and experimentally from delayed
"fluorescence? What evidence is there that the triplet state is
involved in phosphorescence? Discuss the reason why it is
appropriate to think of the ISC as occuring at the intersection
of the two potential curves.
Further information. See MQM Chapter 10, "fluorescence
and laser action. See also Bo wen (1946), Wayne (1970),
Calvert and Pitts (1966), and McGlynn, Azumi, and Kinoshita
(1969). The electron spin resonance evidence for the role of
the triplet state will be found in Hutchison and Mangum
(1958) and Carrington and McLachlan (1967), and is reviewed
in detail in McGlynn eta/. (1969).
photoelectric effect. When shortwavelength light falls on a
metal surface electrons are emitted. Three observations are
very important:
{1} the emission occurs only if the wavelength of the incident
light is smaller than a threshold value characteristic of the
metal;
(2) emission occurs even at very low intensities so long as the
threshold frequency is exceeded, and however dim the light
there is no timedelay between its application and the appear
ance of photoelectrons;
(3) the kinetic energy of the emitted electrons depends
linearly on the frequency of the light once the threshold is
exceeded.
The first observation suggests that the energy that can be
transferred to the meta! surface in order to eject an electron
is related to the frequency of the light, and that the metal
surface cannot gradually accumulate energy until it has
sufficient for the job. This behaviour is nonclassical, because
an incident wave would be expected to deposit its energy
into the metal irrespective of its frequency.
The quantum theory and its concept of "photons can
explain all the features of the effect in a simple and direct
fashion. It recognizes that a light wave of frequency V may be
considered as composed of a collection of photons each
bearing the energy hv. The explanation of the three obser
vations is then as follows.
1. When a photon strikes the metal's surface it can eject an
electron by imparting all its energy to it, but the ejection will
be successful only if the energy transferred hv is sufficient to
overcome the energy that binds the electron to the metal (the
"work function <p). If the frequency is less than the work
function the electron will not be emitted, and the photon re
emerges from the surface as part of the reflected beam. Thus
the threshold frequency of the photoelectric effect can be
understood.
2. The intensity and time characteristics are explained on
the same basis, because so long as the frequency exceeds
threshold the photon is able to eject an electron; the effect
depends on single photonelectron collision events rather than
the accumulation of energy from a passing wavefront . At low
intensities {few photons) only a few collisions occur, but each
photon carries the same energy hu as the photons in a heavily
populated intense beam of the same frequency.
3. The third point can be explained on the grounds that a
successful photon is annihilated in the collision that ejects
the electron, and, by the conservation of energy, all its
energy must appear in the electron; of this energy an
amount corresponding to the work function is expended in
prising the electron out of the metal and setting it in free space,
and the remainder hv — <p must be ascribed to the kinetic
energy of the electron ?r> B v z . It follows that the kinetic
energy of the electron is proportional to the frequency of the
incident light.
The importance of the photoelectric effect, other than its
technological value for lightsensitive devices, lies in its
historical value in the development of the idea that radiation
is "quantized and in its application to the study of the "work
function, for the latter can be determined from the threshold
frequency. A modern development of this is photoetectron
spectroscopy (PES), where electrons are ejected from molecules
by a highenergy photon {from a shortwavelength source). The
'work function' in this case is the energy required to extract
the electron from the orbital it occupies in the molecule, and
so the appearance of electrons with a variety of different
kinetic energies under the influence of monochromatic
radiation implies that they are being ejected from a corre
sponding range of orbitals of various binding energies
{Fig. P7). Analysis of the kineticenergy spectrum of the
photons
177
FIG, P7. Formation of a photoelectron spectrum.
ejected electrons provides detailed information about the
energy levels of molecules. When the photon source is in the
Xray region each photon carries sufficient energy to eject
electrons from the innermost shells of atoms, and so their
energies can also be studied (Fig. P8). This Xray technique is
known by the inglorious name of electron spectroscopy for
chemical analysis (ESC A). See also "Auger effect.
Questions. State the significant features of the photoelectric
effect. Why cannot the classical wave theory of light account
for these phenomena? What is the role of photons in the
photoelectric effect, and how do they enable the features of
the effect to be explained? What is the role of the work func
tion? Light of wavelength 750 nm, 500 nm, 200 nm falls on
a surface composed of one of the following metals: Na (2'3
eV), K (22 eV), Cs (21 eV), W (45 eV); the number in
brackets is the work function for the metal. Predict the
O K tt (54keV)
Al K a (.49 IteV)—
Y M 5 CU2eV)
He li(4leV)
He IC2I«V)
FIG. P8. Ejection of electrons from different regions of an atom.
kinetic energy of the ejected electron in each case where
photoe mission may occur. The kinetic energy of photo
electrons as a function of incident frequency was measured: at
a wavelength of 625 nm the kinetic energy was 02 eV, at
416 nm it was 12eV, and at 312 nm it was 22 eV, Calculate
the work function for the metal. What is the relation of
photoelectron spectroscopy to the photoelectric effect?
Further information. Analysis of the photoelectric effect will
be found in Chapter 2 of Bohm (1951) and 81.3 of Jammer
(1966). The implications of the phenomenon for the concept
of photons is described by Born and Beim (1968). Photo
electron spectroscopy is described in Turner, Baker, and
Brundle (1970), Baker and Betteridge (1972), and Siegbahn
etal. (1969). Siegbahn (1973) has given a simple introduction
to ESCA and Turner (1968) a simple introduction to PES.
photons. Light of frequency V can impart energy only in dis
crete amounts ("quanta) of magnitude hv, where h is Planck's
constant. Alight beam of frequency v therefore can possess an
energy that is an integral multiple of hv, and so it is natural to
178
polarizability
imagine this successive excitation of a frequency mode as an
addition of hypothetical particles to a state. Thus a beam of
frequency fand energy nh V could be regarded as containing n
light corpuscles. These quanta of excitation, or hypothetical
corpuscles, are photons. Each photon of wavelength A = civ
carries an energy hv and, according to the *de Brogiie relation,
a linear momentum hvlc. Lowfrequency photons carry little
energy and momentum; highfrequency photons carry much
of both.
Does light exhibit any of the corpuscular features that this
photon description suggests? Indeed it does, for the "photo
electric effect confirms that energy can be transferred only in
discrete amounts (corresponding to the annihilation of a
photon), and the "Compton effect shows that each photon
carries a characteristic amount of momentum related to its
frequency. Radiation pressure can be understood in terms of
the pressure imparted to a surface by a steady stream of
incident photons.
Photons may be polarized so that their electric component
lies in a plane (planepolarized light) or moves in a circle or
ellipse; but in all cases the electric component must be
perpendicular to the propagation direction. The photon
possesses an intrinsic angular momentum (its "spin), and of
this the existence of left and right circularly polarized light
is a manifestation. The spin of a photon is unity: one pro
jection (that corresponding to a righthand screw along the
direction of travel) corresponds to left circularly polarized
light, one projection (the lefthand screw) to right circularly
polarized light, and the third (remember that for "spin 1
there are three possible components on a selected axis) is
forbidden to particles moving with the speed of light, which
photons do.
The existence of the spin of a photon is the basis of the
^selection rules for "electric dipole transitions. Since its spin
is unity the photon is a "boson: this entails, through the
•Pauli principle, that an indefinite number may occupy a
single quantum state, and for this reason intense mono
chromatic beams may be prepared {see "laser).
Questions. What is meant by the term "photon? What is the
energy of a photon in light of frequency v, and what
momentum does it carry? What is the energy of the photon
of light corresponding to the microwave region of the
spectrum (1 cm"') and infrared (500 enf '), the visible
(~55Q nm), the ultraviolet (300 nm), and 7rays (10"" m)?
How many photons are emitted in a 1 mJ pulse from a 337
nm wavelength nitrogengas laser? How many photons are
emitted from a tungsten 100 W lamp each second: assume all
the radiation is at 450 nm? What frequency of light is the
minimum that can be used to fracture a 100 kJ mof 1 bond
in a molecule? What is the evidence for the corpuscular
nature of light? Why is this view compatible with the exist
ence of the typically wave phenomenon of diffraction?
Further information. The quantum mechanics of photons
can be made very complicated; but it is very important. now
that lasers are ascendant. If you are really interested see
Grandy (1970), Louisell (1973), Berestetskii, Lifshitz, and
Pitaevskii (1971), Akhiezer and Berestetskii (1965), Levich,
Mayamlin, and Vdovin (1973), Jauch and Rohrlich (1955),
and Kaempffer (1965). The uncertainty relation for the
phase of photons and their number is discussed correctly by
Carruthers and Nieto (1968), but rarely correctly elsewhere.
The study of photons is helpful in the discussion of the inter
pretation of quantum mechanics: see Dirac (1958) and
Feynman, Letghton, and Sands (1963). Quantum optics is
the study of optics where the quantum, and therefore the
photon, aspects are dominant. See Glauber (1969) for a
review and Loudon (1974) and Louisell (1973) for a modern
account. Are photons particles? See Born and Beim (1968) for
a view. See p. 9 of Whittaker's Stereochemistry and mechanism
(OCS 5) for a helpful picture of the decomposition of a plane
wave into its circular components.
polarizability. When an electric field is applied to an
individual atom or molecule the electron distribution and the
molecular geometry are distorted; the polarizability is a
measure of the ease with which this occurs. The atomic
polarizability is the contribution to the overall polarizability
due to the geometrical distortion. It is usually significantly
smaller than the electronic polarizability , which is the con
tribution due to the displacement of the electrons. There is a third
contribution to the polarizability of a bulk sample: this is the
orientation polarization. It arises when the molecules have a
polarizability
179
BOX 17: Polarizability
Energy of potarizable molecules in a field E
£(E)=£<°> // >E ^ lj.^3 + . . .
Dipole moment in a field E
M is the permanent dipole moment, or the polarizability,
and the first hyperpolarizability. (For magnetic properties,
a is called the magnetizability, and written £.)
Quantum expressions
a = (<y£/9E% s0 = § £Vd ,,d n0 .
A„ = f^ 0) — E< 0) , the molecular excitation energy,
d on = ^ ldli^ n ), the transition dipole moment.
Polarization of medium
Electric susceptibility ()^): P = e % E.
Relative permittivity {dielectric constant): € = t + v.
Refractive index: n = e /a
r r
Lorentz local field: E qc  E + P/3e = (e + 2)E
(the factor 3 is approximate).
Lorentz Lorenz or ClausiusMossotti equation:
( e r 1 \ (m\ _ La
M is the molecular weight, p is the density, and L is
Avogadro's number. La/3e is referred to as the molar
polarizability a
M'
Debye equation (for molar polarizability of polar
dielectrics):
a A : atomic polarizability
a E : electronic (molecular) polarizability
a Q : orientation polarizability, which is given by the
Langevin contribution for a molecule with
permanent dipole moment uS a \ as
0L o =l± W2 l3kT,
See Table 20 for the Maxwell equations.
permanent dipole moment: the applied field orientates the
molecules, and the entire sample acquires net polarization. The
orientation effect is not complete because thermal motion
disorganizes the sample: its magnitude may be calculated by
applying the Boltzmann distribution to determine the mean
dipole moment of the sample, and one deduces the Langevin
contribution exhibited in Box 17. This mechanical con
tribution will not concern us further (although it is an
important component of the total polarization); we shall
concentrate on the source of the electronic contribution.
Highly polarizable molecules respond strongly to the
application of the field; they become highly polarized, and the
centroid of negative electronic charge is displaced. If the mole
cule is initially nonpolar the polarization leads to the form
ation of an electric "dipole moment; and if it is already polar
it leads to an additional component of the dipole. The
magnitude of the induced dipole is a good indication of the
polarizability of the molecule, and the polarizability a may
be defined as the constant of proportionality between the
induced moment and the strength of the electric field:
u(induced) =aE. The dipole moment mighfdepart from this
linear relation if the applied field is very strong; in that case
the dipole depends on E 2 , and higher powers, and the coef
ficients of E 2 , F 3 , . , . are known as the first, second, . . .
hyperpolarizabilities. We shall neglect these nonlinear response
terms and concentrate on the linear response, the polarizability
a.
The quantummechanical calculation of the polarizability
proceeds by calculating the energy of a polarizable molecule in
an electric field, and relating this to a secondorder
"perturbationtheory calculation (see Box 17). It is found that
the magnitude of polarizability can be interpreted in a variety
of ways.
180
polaron
1. One interpretation shows that the polarizability increases
with the size of the atom and with the number of electrons it
contains; this can be understood in terms of it being easier for
a field to distort the electronic distribution when the electrons
are far from the nucleus, or well shielded from its charge.
2. Another interpretation, which is based on the view that
the distortion of the molecule can be represented by an
appropriate "superposition of wavefunctions, gives the
expression for the polarizability as a sum of terms, each one of
which represents an admixture of an excited state into the
ground state. The amount of each state depends on its energy
separation from the ground state and on the intensity of an
optical, electric dipole transition between it and the ground
State: as the intensity of the transition increases the state may
contribute more strongly, and as the energy increases it con
tributes less. A consequence is that it is reasonable to expect
molecules that have intense transitions in the optical or lower
frequency region of the spectrum to be highly polarizable. On
this basts it is understandable that the aliphatic hydrocarbons,
which have weak optical transitions in the ultraviolet, are only
weakly polarizable.
3. A third interpretation remarks that the polarizability
formula can be understood in terms of the magnitude of the
fluctuations in the instantaneous electric dipole moment of
the species. It is possible to imagine local transient electronic
movements in the molecule that give rise to a dipole moment
which on the average is zero (for a non polar molecule); the
greater these fluctuations the greater the polarizability. This
interpretation is related to the first, because the fluctuations
are greater in large, weaklybound systems.
The polarizability of a sample is frequencydependent. At
low frequencies (below about 10 12 Hz) the molecules, the
atoms within the molecules, and the electrons of the molecules
can follow the changing direction of the applied field. At
higher frequencies (above 10 12 Hz but below 10 14 Hz) the
molecules cannot reorientate themselves sufficiently quickly,
and so the orientation polarizability ceases to contribute. At
still higher frequencies the atomic nuclei are too sluggish to
follow the oscillating field, and the atomic contribution is
eliminated. This implies that at high frequencies (in the
optical range) the only contribution to the polarizability is the
electronic, but even this ceases at very high frequencies. The
frequencydependence (the dispersion) of the polarizability is
a helpful property in distinguishing the contributions.
Polarizabilities of molecules are related to the relative
permittivity (dielectric constant) of the medium they compose
{see Box 17), and its "refractive index. Both properties enable
the polarizability to be determined over a wide frequency
range. (Since the refractive index is normally measured at
optical frequencies it is related to the electronic polarizability.)
Questions. 1. What happens when a molecule is immersed in
an electric field? How may the polarizability be defined? What
is the dependence of the induced dipole moment on the
applied field when the latter is very strong? Would you expect
the polarizability of a molecule to depend on its orientation
with respect to the field? Why should the polarizability
increase as the species becomes larger and its electrons less
strongly bound? Which is more polarizable, He or He + ? What
is the role of the instantaneous fluctuations of the dipole
moment of an atom in determining its polarizability? Can the
polarizability be related to the strength and frequency of
optical transitions? Why does the polarizability depend on the
admixture of excited states? {Revise perturbation theory and
■virtual transitions.)
2. Apply the expression in Box 17 to calculate the polariz
ability of a charged simple harmonic oscillator, an electron in
a onedimensional square well, and the hydrogen atom. Where
you feel it necessary apply the closure approximation. Cal
culate the polarizability parallel and perpendicular to the axis
of a 2p orbital in the carbon atom: use "Slater atomic
orbitals and the closure approximation.
Further information. See MQM Chapter 1 1 for a derivation of
the relations in the Box and for a further discussion. See also
van Vleck (1932), Davies (1967), and Buckingham {1960).
Polarizabilities are listed in LandoltBornstein. For further
applications see intermolecular forces and "dispersion
forces. Hyperpolarizabilities and their measurement are dis
cussed by Buckingham and Orr (1967). See also Kielich
(1970).
polaron. A polaron is a defect in an ionic crystal that is
formed when an excess of charge at a particular point
polarizes the lattice in its vicinity. Thus if an electron is
precession
181
OOOOCDOO
OOOOCD0O
eeeee
eeee
ooo
eeee
eeee
cpoooc
ooooc
eeee^
ee^
oooc
»eee£
OOOOMOWOOOC
000000000000c
FIG. P9. A polaron in a simple lattice. The doubly hatched site is a
doublv charged region; the singly hatched sites are singly charged
with opposite sign.
captured by a halide ton in an alkalihalide crystal the metal
ions move towards it and the other negative ions shrink away
(Fig. P9). As the electron moves through the lattice it is
accompanied by this distortion. Dragging this distortion around
effectively makes the electron into a more massive particle,
and this is intended to be implied by the name polaron: a
lattice distortion moving through the lattice like a massive
particle.
Further information. See Chapter 10 of Kittel (1971) and
Kittel (1963). A good discussion is given in Chapter 4 of Mott
and Davis (1971 ), and the subject is reviewed in Kuper and
Whitfield (1963).
precession. In the vector model of the atom and in the
general theory of angular momentum, an angular momentum
of magnitude [/{/ + 1)] y 'h andzcomponent/nfi is represented
by a vector of length [/"(/ + 1 )] K making a projection m on to
azaxis. From the uncertainty principle it is known that if the
^component of angular momentum is precisely specified then
the x and the ycomponents are completely uncertain. This
situation may be represented in the vector diagram by
indicating the range of possible orientations of the angular
FIG. P10. Precession. The cone denotes the possible but indeterminate
orientations of the vector.
momentum by a cone (Fig. P1Q, but see also Figs. A2 and A3):
wherever the vector lies on this cone, it has the samez
component but its* and ycomponents are completely
undetermined. The cone is referred to as the cone of
precession. Note that we avoided saying that the angular
momentum actively precesses around z: the cone represents
the possible array of orientations of the angular momentum,
and at this stage we do not wish to imply that the tip of the
vector moves round the mouth of the cone. In the absence of
magnetic fields the vector is at rest at an indeterminate position
on the conical surface.
When a magnetic field ts applied along the zaxis the states
with different projection m have different energies by virtue of
the magnetic moment associated with the angular momentum:
the energy of the state m ism^ a S for orbital angular
momentum (and 2m(i B B for »spin}. This energy can be
expressed as a frequency by dividing by h: CU= mft B/ti The
vector diagram is a symbolic code representing the quantum
mechanics of the situation, and it can be augmented in a way
that incorporates the energy of the state by adopting the
convention that a state with energy mp. B is represented by a
vector that revolves around the zaxis with a frequency 03
(Fig. P1 1 ) : this is precession, and the frequency is the Larmor
precession frequency. As the field is made weaker the
precession frequency slows, and in the limit of zero field the
static, indeterminate distribition of vecots is regained. The
state with the greatest value of \tn I precesses most rapidly, and
182
precession
fairer
still
falter
=/\B/fi
cj0
Pffr
FIG, Pit. Larmor preeBssion in the presence of a magnetic field, and its
connexion with the vector model,
that with zerom precesses not at all; opposite signs of m are
interpreted as opposite senses of precession.
From this picture we see that the rate of precession about
an axis represents the strength of coupling to that axis, and
this view may be extended to situations where the energy
arises from sources other than external magnetic fields. The
FIG. P12. Two points of view about the precession of coupled vectors.
In (a) we sit on I and observe s and j; in (bl we sit on j and observe 5
and I.
case of °spinorbit coupling is an example: I and s are coupled
by the spinorbit interaction, and form a vector j. If the spin
orbit coupling is strong the vectorcoupling scheme involves a
rapid precession of s about 1 (or, what is equivalent, a rapid
precession of 1 and s about their resultant j): this is illustrated
in Fig. P12. When the coupling is weak the precession is slow
and the coupling can easily be broken by other influences.
The picture of precession is a superb example of the way
that algebraic concepts of quantum mechanics can be
represented by diagrams: each aspect of the 'picture' is a code
for some aspect of the quantummechanical situation. But
then, you might think, is the quantummechanical algebra
not itself merely a code?
Questions. What does the cone of precession represent in the
vector model of angular momentum when no field is present?
Why is a cone forced on the model? What does it represent
when a magnetic field is applied? Upon what does the
precession frequency depend? Calculate the Larmor
precession frequencies for a pelectron in a magnetic field of
1 G and 3 kG; what is the Larmor frequency for an electron
spin in a field of 34 kG and of a proton spin in a field of
15 kG? What in general does the rate of precession represent?
Is it easy to decouple two momenta that are rapidly precessing
about each other? The spinorbit coupling energy of an elec
tron in a firstrow atom is about 50 cm 1 : compute the
relative precession frequency for this situation, and estimate
the external magnetic field that would be required to cause a
significant decoupling of the momenta and a significant
orientation of the individual momenta with the field. The
conversion of a triplet state into a "singlet can be brought
about by changing the relative phase of the two electronspin
orientations: thus if one spin precesses faster than the other
its orientation is shifted by 180° with respect to the other, and
a singlet is generated out of a triplet (see singlets and triplets,
especially Fig. S7). Suppose that the two electrons of a triplet
molecule are in fields differing by 1 G, either by virtue of
inhomogeneities in the field or because they experience
different internal fields. Calculate, by determining the different
Larmor precession frequencies, the rate at which the singlet
is formed. Can you see a connexion with the interconversion
of ortho and parahydrogen (see "nuclear statistics)?
predissociation
183
Further information. See MQM Chapters 6 and 8 for a dis
cussion of the algebra beneath the picture, and the applications
of the "vector model to atomic and molecular spectra. See
also Candler (1964), Herzberg (1944), White (1934}, and Kuhn
(1962) for applications in atomic spectroscopy. The language of
precession is used extensively in the discussion of magnetic
resonance experiments; therefore see McLaucfolan's Magnetic
resonance (OCS 1), LyndenBell and Harris (1969), Carrington
and McLachlan (1967), and Slichter (1963). A good example
of the decoupling of two precessing vectors is provided by the
°PaschenBack effect.
predissociation. Ordinary, wellbehaved dissociation occurs
when a molecule is excited to a state that possesses more
energy than the separated fragments, A transition from curve
X to curve A in Fig. P13 a provides such an example; the
spectrum is blurred where the upper vibrational states are not
•quantized (where they are really translations! states}. Pre
dissociation is dissociation that occurs in a transition before
the dissociation limit is attained, hence its name.
them to remain until enough energy is added to excite'the
molecule beyond the dissociation limit of the upper state. In
Fig. P13 b we see that the vibrational structure disappears at
this point. (Below this point it has the intensities characteristic
of the "selection rules in operation and the °Franck : Condon
principle.} In some cases it is observed that the vibrational
structure of the spectrum disappears before the dissociation
limit is attained. This may occur because another, dissociative
state (B) crosses A, and because there exists an interaction
("perturbation) in the molecule that can flip the state of the
excited molecule from A to B (Fig. P13 c). Put another way,
this can be interpreted as the states in the vicinity of the
region A in Fig, P13 c being a mixture: the vibrational states of
A acquire some of the translational character of the states of B
at that energy. Therefore a state within the region A has some
propensity to dissociate even though its energy may be far
below the dissociation energy for the state A. When the energy
of the incident light is great enough to excite the vibrational
state of A above the region A the lines are again observed to be
sharp, for now the molecule is unable to switch into the state
dissociation limit
FIG. P13. (a) Dissociation by a transition to a dissociative state, (b) Dissociation when enough energy is added to disrupt the upper state, (c) Pre
dissociation due to the presence of a dissociative state. The appearances of the spectra are indicated on the left.
Consider the transitions A *— X shown in Fig. P13 b. On
the electronic transition is superimposed a series of lines due
to the transition to the vibrational states of A. At low
frequencies (energies) the lines are sharp, and so we expect
B, The presence of predissociation may be recognized therefore
by a blurred region in the vibrational progression of an elec
tronic transition.
The mechanism of predissociation is closely related to the
184
progression
•Auger effect, the principal difference being that the former
dissociates a fragment of the molecule and the latter spits out
an electron. In each there is a radiationless transition from a
bound to an unbound state. Predissociation obeys certain
•selection rules, and these will be found listed in that section.
Induced predissociation is predissociation that is induced
by some external influence; in particular, collisions with a
foreign gas {'collisioninduced predissociation') or an applied
field. The collisions are able to knock the excited molecule
from A to B, and the applied field may relax some of the
selection rules that govern the AvWV* B transition (the wavy
line denotes that the transition is radiationless).
Questions. 1. What is meant by the term 'predissociation'?
How may it be recognized in a spectrum? Suppose the
emission of light from a molecule were observed: what effect
would predissociation have on the appearance of the emission
spectrum? What causes the molecule to flip from one curve
to another? How may the efficiency of this transition be
enhanced? In what senses is predissociation a type of "Auger
effect? What happens to the rotational structure of the
electronic transition in the region close to the predissociation
domain? Suppose that the state dissociates with firstorder
kinetics and lifetime r, use the "uncertainty principle to find
an expression in terms of r which determines whether or not
rotational structure will disappear from the spectrum, and then
find an expression for the disappearance of vibrational struc
ture. Choose typical values of the molecular properties you
require, and assess the lifetimes of levels for which the
rotational and vibrational structures disappear.
2. In the text it was remarked that outside the region A
there was only insignificant mixing of the states A and B. On
what grounds may that statement be justified? Provide a
classical and a quantummechanical interpretation; for the
latter consider the role of overlap in the same way as in the
justification of the "FranckCondon principle.
Further information. See MQM Chapter 10, See Herzberg
[1950, 1966) for a thorough discussion of predissociation and
many examples. See also Barrow (1962), Gaydon (1968), and
King (1964). Induced predissociation is described by Wayne
(1970).
progression. In the "electronic spectra of molecules the
excitation of the electron is accompanied by excitation of the
"vibrations, and so instead of a single line in the spectrum there
may be a complicated band of transitions. A progression is a
series of lines that arise from transitions from the same
vibrational level of one of the states (the ground electronic
state if absorption is involved) to successive vibrational levels
of the other state. Thus the v" = progression is the series of
transitions starting in the v" = vibrational level of the ground
electronic state of the molecule and terminating in the V = 0,
1,2,... vibrational levels of the upper electronic state (Fig.
P14). The v" = 1 progression is a similar series starting in the
(/' = 1 vibrational level of the ground electronic state. The
lines in a progression are labelled (v' , v"); therefore the
if = progression consists of the transitions (0, 0), (1, 0),
{2, 0), etc..
+
■ ,
J
.'
I
J
,
C0,0KI,0>«,Q>
5 U
4
3
2
I
6 u"
5
4
3
FIG. P14. A progression.
Questions. What is a progression? How does it differ from a
"sequence? What information can you extract from the
positions of the lines in a progression, and what information
can you obtain fay comparing the v" = and the v" = 1
progressions? Would you expect the intensities of the pro
gressions to be the same? Given that the energy of a
vibrational level of the ground state depends on v" through the
progression 185
expansion iJ'{u" + ~) + x"oJ'{if + ) 2 , with a similar
expression for the upper state, calculate the frequencies of the
if — and the v = 1 progressions.
Further information. See MQM Chapter 10. Detailed
information about the appearance, analysis, and formation
of progressions will be found in Barrow (1962), Whiffen
(1972), Gaydon (1968), King (1964), and Herzberg (1950).
Q
quadru poles and other multipoles. The electric quadrupole
moment is one of the series of rnultipole moments which are
used to describe the way electric charge is distributed over a
body. The first member of the series is the electric monopole,
more commonly referred to as the point charge; then comes
the electric 'dipole moment, which may be regarded as the
juxtaposition of two opposite charges, and so has no net charge
(no monopole moment); then comes the electric quadrupole
which may be regarded as being formed from four charges
arranged in a way that leads neither to net charge (no monopole
octopole
FIG. Q1 . Various electric rnu I tipoles.
hexodecapole
moment) nor net dipole moment. An arrangement that
achieves this is shown in Fig. Q1 . Higher multipoles may be
constructed in an analogous way: for a 2 n pole {n = 1 is a
dipole, n  2 a quadrupole, n = 3 an octopole, n = 4 a
hexadecapole) it is necessary to arrange 2" electric charges in
an array that possesses no lower rnultipole. Some of these are
represented in Fig. Q1, but alternative arrangements may also
be envisaged.
Often many multipoles vanish by virtue of the symmetry of
the molecule. Consider, for example, the molecule CO2 which
has the linear structure 0=C=0. As the molecule has no net
charge, it has no electric monopole moment. It is symmetrical
about the carbon atom, and so it has no electric dipole moment.
The oxygen atoms are more "electronegative than the carbon,
and so they bear a higher charge density than the central
carbon; therefore the charge distribution has the form
(5 —)—{S + 8 +)(S ). This is of the form of a quadrupole,
and so we can expect the quadrupole moment of CO^ to be
nonzero, as indeed is found.
The potential arising from a general charge distribution may
be represented by a sum of potentials of the multipoles into
which the distribution may be divided. The electric potential
due to a 2 n pole falls off with distance according to 1/r" +l ;
therefore the electric field due to a 2" pole falls off
according to l/r n * 2 (the field is the negative gradient of the
potential). It follows that the higher multipoles have a much
shorter range than the lower: a monopole potential falls off
according to the Coulomb law 1 lr, a quadrupole according to
I// 3 . The more rapid decay of potential in the higher
multipoles may be understood in terms of the way that a
186
cluster of electric charges (as depicted in Fig, Q1 ), when
viewed from great distance, is hardly distinguishable from
having nothing at all: the cancellation at large distances occurs
more effectively the higher the rank of the multipole.
quadruples and other multipoles
187
BOX 18: Multipole fields
Field at r due to a point multipole at origin
£ n *1/r"* a .
Potential at r
<x Mr"* 1
dip0le: \4neJ {?) cos B <9. ^ separated by R, n = qR).
linear quadruple: j(ff^) (^J (3cos 2 01)
(quadrupole formed with ju, fi headtohead, eQ = 2o7? 2 ).
Energy of multipole in a field E
(1) point charge (monopoie) £ = q(p[r), where
(2) dipole: &<•*%£
(3) quadrupole: £ = $eQ{d 2 <t>/dz 2 )[3 cos 2 01).
J
Different multipoles interact with different features of the
electric field. An electric monopote interacts with the electric
potential itself (see Box 18). An electric dipole interacts not
with the potential but with its gradient; that is, the electric
dipole interacts with the electric field. This can be understood
in terms of the structure of the dipole as two juxtaposed
point charges: at a general orientation of the dipole one charge
interacts with the potential at its position and the other
opposite charge interacts with the potential at its position.
Only if the potential is different at the two points is there a
net interaction; therefore there is an interaction only if the
gradient of the potential does not vanish at the position of
the dipole. Those to whom extension of analogies gives
pleasure will be pleased to discover that the electric quad
mpole moment interacts with the second derivative of the
electric potential; or, what is the same thing, with the gradient
of the electric field. This may be understood by considering
the situations depicted in Fig. Q2. In the first, where the
gradient of the potential (the field) is constant over the
quadrupole, the energy of interaction is independent of
orientation, and so there is no net interaction. In the second,
where the potential has curvature (and where the field has a
gradient), the energy depends on the orientation, and
therefore we conclude that there is an interaction. The
mathematical form of the interaction is given in Box 18.
r^)
>f
ff'f}
.'field
/
FIG. Q2. (a) A quadmpole in a constant field has an energy indepen
dent of orientation, (b) A quadmpole in a field gradient has an energy
dependent on orientation. In each case the density of shading denotes
the strength of ihe potential.
The typical magnitude of a molecular mono pole is of the
order of the magnitude of the electronic charge e (1.6 X 10" 19 C).
A single charged ion may be regarded, by the pedantic or the
neat, as a monopofe of this magnitude. The magnitude of a
molecular dipole is of the order of eR, where R is a molecular
diameter; for R  0.1 nm (1 A) the order of magnitude is IfX 29
188
quantum
C m (but see °dipole moment for debye). In the same way we
may estimate the order of magnitude of the electric quadrupole
moment of a molecule as eR 1 , or roughly 10" 3 * C m 2 . Part of
the absurdity of the size of the unit is eliminated by defining
the electric quadrupole moment as eQ, and calling Q the
'quadrupole moment'. It is normally expressed in square
centimetres; therefore a molecular quadrupole moment might
be expected to be of the order of 10~ 16 cm 2 (1 A 2 ), and a
nuclear quadrupole moment (see hyperfine effects) of the
order of 10" 24 cm 2 .
Questions. 1. Draw diagrams of charge distributions that
represent an electric quadrupole, octopole, hexadecapole,
32pole. Now draw alternative structures with similar
multipoles. What is the dependence of distance of the
potential due to a 2"pole? What is the dependence on distance
of the field due to a T pole? Why is the electric quadrupole
moment the most important multipole moment of COj?
What is the first nonvanishing multipole moment of the
helium atom? Of methane? Of water? Why does the potential
of a quadrupole fall off more rapidly that that of a dipole?
With what aspect of an electric field does a quadrupole interact?
Why does a quadrupole not interact with a constant electric
field? With what aspect of the field does an electric octopole
interact? Draw a diagram to illustrate the physical basis of
your answer. How may a molecular quadrupole be measured?
2. Take the quadrupole arrays of charges illustrated in Fig. Q1
and consider a point at a distance r from the centre of both
arrays. Write an expression for the potential at that point due
to the multipole (by regarding each point of the multipole as
the source of a Coulomb field). Assume that r is much greater
than the separation tfof charges within the multipole, and
expand the potential in powers of d/r; retain the leading term
in the expansion. In this way find the electric potential due to
a quadrupole. Then find the field at the point r. Compare your
answer with the expression in Box 18.
3. Put a single electron at a distance r from the centre of a
quadrupole of (a) molecular dimension and (b) nuclear
dimension. Calculate the energy needed to rotate the quad
rupole through 90°. What size quantum would be needed to
invert a nucleus of this nature? To what frequency does that
correspond? This calculation is the basis of the technique of
nuclear quadrupole resonance (n.q.r.), where electric field
gradients within molecules are determined by observing the
energy required to rotate the orientations of nuclei with
electric quadrupole moments.
Further information. Simple calculations on the electrostatics
of multipole moments will be found in Corson and Lorrain
(1970); tougher accounts are given by Jackson (1962) and Rose
(1955, 1957). See Sugden and Kenney (1965) for a simple
account with special attention paid to the way that quadrupole
moments play a part in molecular spectroscopy. The calculation
of molecular quadrupole moments is described by Davies (1967)
and their measurement by Buckingham (1965), who also
considers their rote in intermolecular forces. The determination
of nuclear quadrupole moments is described by Lucken (1969),
who also gives a detailed account of nuclear quadrupole
resonance spectroscopy; see also Das and Hahn (1958). A
recent review of n.q.r. is that by Chihara and Nakamara (1972).
See Carrington and McLachlan (1967) for a simple account of
the effects of quadrupole moments in magnetic resonance,
and Chapter 6 of Slichter (1963) for a more sophisticated, but
well stated, account. Tables of nuclear quadrupole moments
are given in §8 of Gray (1972), and a few will be found in
Table 17.
quantum. A quantum of energy is the smallest amount of
energy that may be transferred to or from a system. In
classical physics there is no limitation to the smallness of
permissible energy changes of a system, but quantum physics
shows that only quantities of energy of a welldefined size
can be transferred, the actual size depending on both the
system and its state. A particularly simple example is provided
by the harmonic oscillator of natural period v. In classical
theory the oscillator may swing at its natural frequency with
some energy, and its energy may be changed continuously.
Quantum theory shows that the oscillator's energy can change
only by some integral multiple of hv, where h is Planck's
constant. Thus an attempt to change its energy by a fraction
of hv must fail.
The magnitude of quanta involved in the behaviour of
macroscopic objects is so small that the changes in energy are
virtually continuous, but at a microscopic level the quan
quantum defect
189
tization cannot be ignored. For example, the size of the
quantum needed to transfer energy to and from the pendulum
of a grandfather clock {v~ 0.5 Hz) is only 3 X 10~ 34 J; but
the oscillation of a bond in a molecule occurs at a frequency
of the order of 10' 4 times larger at 6 X 10~ 20 J, or 36 kJ mol 1
When an atom falls from an excited state of energy £ the
change of energy f (  E g is emitted as a quantum of light with
a frequency V given by the Bohr frequency condition hv =
E e  f g The size of the emitted quantum depends on the
states involved in the transition: quantum changes of large
energy appear as highfrequency radiation, and those of small
energy as lowfrequency radiation.
Planck's constant itself has the dimensions of action (J s)
and so may be regarded as the fundamental quantum of action.
The early approach to quantum theory proceeded by limiting
the action of a system to multiples of h, and the
BohrSommerfeld quantization condition, which was the
basis of the old quantum theory, was based on this
quantization scheme. The dimensions of angular momentum
are the same as those of action, and the natural quantum of
angular momentum is also Pfanck's constant: the angular
momentum of a rotating body may be changed only in
amounts of the order of h (h = h/2ir is a better order of
magnitude), and so a rotating wheel cannot be decelerated
continuously and smoothly, but must deliver up its angular
momentum (and its energy) in steps. Once again the angular
momentum quanta are so minute for macroscopic bodies that
it can be varied virtually continuously, but on an atomic scale
the effects of quantization are of profound importance.
Questions. What is a quantum of energy? Discuss the excitation
of a simple pendulum in both classical and quantum terms.
How is the energy of the quantum associated with the
excitation of an oscillator of frequency Vl Calculate the size
of the energy quantum for processes of natural frequencies
1 Hz, t0 10 Hz, and 10 1S Hz. Calculate the energy of a
quantum of red, yellow, and blue light, of ultraviolet light,
and of Xrays (get help from Table 5).
Further information. See MQM Chapter 1 for an historical
account of the realization of the necessity of quantization,
and Chapter 3 for some of its simpler manifestations. See
Jammer (1966) for an historical perspective, and Hoffmann
(1959), Heisenberg (1930), and Andrade e Silva and Lochak
(1969) for gentle accounts. The experimental evidence is also
reviewed In MoelwynHughes (1961), Slater (1968), and Bohm
(1951). See Feynman, Leighton, and Sands (1963) for a
thorough and illuminating discussion of the whole subject.
The Josephson effect provides an excellent method for
measuring h\ for a description of the effect, see Langenberg,
Scalapino, and Taylor (1966), and for its application, see
Taylor, Langenberg, and Parker (1970).
quantum defect. The spectrum of atomic "hydrogen
(which is illustrated in Fig. H10 on p. 104) consists of
several "series of lines which can be represented as the
difference of two terms, both of which have the form
Bin 1 , where R is the Rydberg constant and n is an integer.
The spectra of the alkati metals may also be grouped into
series that can be expressed as the difference of two terms of
the form R'/n 2 ; but on account of the repulsive effect of
the core electrons the number n, the effective quantum
number, is no longer an integer. It may be written as
n — 5, where n is an integer and 5 is a correction called the
quantum defect. This defect diminishes as the principal
quantum number of the electron increases, for the electron
is further away and the nucleus and its surrounding core
electrons resemble quite accurately a single positive point
charge, and the spectrum and the energies become more
hydrogenlike. Since the quantum defect depends very
strongly on the interaction of the valence electron and the
core electrons it is not surprising that the greatest defects
occur for sorbitals, which "penetrate most closely to the
nucleus. The quantum defect is a guide to the extent of
penetration, but it has little other theoretical significance
or importance.
Further information . The quantum defect occupied more of
the older literature than it does of the modern. Mention of it
will be found in 83.6 of King (1964), §1.5 of Herzberg (1944),
and Chapter III of Kuhn (1962), who lists some values and
discusses why they were of importance. In this connexion,
see §8.5 of Hartree (1957) and S8 of Condon and Shortley
(1963).
190
quantum electrodynamics
quantum electrodynamics. The apotheosis of presentday
quantum mechanics seems to be quantum electrodynamics,
although it is not entirely clear that the theory exists. What
might exist is a unified view of particles and fields in which
the electron is a manifestation of the electromagnetic field.
Like Hamlet's ghost the electron momentarily slips back into
the electromagnetic field, loses corporeality, and then
regroups itself again as a recognizable particle. The electron
spends 1/137 of its existence as radiation {which, incidentally,
is about onequarter of the proportion of the play Hamlet
pere spends as ghost), and the remainder as particle (see
■fine structure constant). This intimate connexion between
matter and radiation is emphasized by quantum electro
dynamics, which treats the radiation field and the particle as
the same object. One consequence of the electron being a
manifestation of the electromagnetic field is that its motion
cannot be smooth on a microscopic scale, but it should be
envisaged as jitterbugging along: Zitterbewegung is the German
with which the idea is elevated into respectability.
For the chemist there are two important manifestations of
this jitterbugging. One is the Lamb shift and the other is the
freespin "0value of the electron. According to the "Dirac
theory of the "hydrogen atom the levels 2 Syi and 2 P]/2 should
be exactly degenerate, but Lamb measured a very small
splitting (of the order of 1057 MHz). Therefore the Dirac
theory, good as it is, must be wrong.
In order to sketch the quantumelectrodynamic explanation
of the Lamb shift we must first establish a crude picture of
what is meant by the vacuum of the electromagnetic field. The
picture of a light beam as a collection of "photons is based on
the view that a mode of frequency V of the electromagnetic
field can be excited to an energy which is an integral multiple
of hv. This is analogous to the way in which a harmonic
oscillator can be excited, and it is tempting to extend the
analogy, and to say that, when no excitation is present (no
photons), the mode still possesses a zeropoint energy. This
zeropoint energy may be traced to the unquenchable zero
point fluctuations of the electric and magnetic fields, just as in
a harmonic oscillator the motion cannot be entirely eliminated.
The field oscillations buffet the electron, and so it jitterbugs
about an equilibrium position. This slight smearing of the pos
ition of the electron affects the energy of an electron in an s
orbital slightly more than one in a porbital because in an sorbital
an electron goes closer to the nucleus and the smearing effect
is more important, and the consequence is that their energies
diverge. This shift may be calculated by quantum electro
dynamics, and almost exact agreement with experiment is
obtained.
The ^value of an electron measures its effective "magnetic
moment arising from its spin. Once again we may envisage an
electron as being buffeted by zeropoint electromagnetic
fields as it spins, and, instead of spinning smoothly, it rocks.
If we imagine a vibration in the equatorial plane the magnetic
moment may be calculated, and once again essentially exact
agreement with experiment is obtained.
Further information. Books on quantum electrodynamics,
disarmingly referred to as QED, are difficult to penetrate.
See G randy (1970), Power (1964), Feynman {1962a, 19626),
Roman (1969) (who gives a helpful guide to further reading),
Bjorkenand Drell (1965), Henley and Thirring (1962),
Thirring (1958), and Schweber (1961). A collection of
significant original papers has been prepared by Sch winger
(1958). The existence of zeropoint fluctuations in the vacuum
is also related to the attractive force that two conducting sheets
exert on each other at small separations: for details of this
CasimirPolder interaction , see p. 142 of G randy (1970) and
§3.4 of Power (1964).
quantum numbers. Quantum numbers are labels that
distinguish the state of a system and, in simple cases, enable
the value of an observable to be calculated. Thus the state of
a "particle in a box is labelled by the quantum number/?, which
may take all integral values greater than zero, and a particle in
a state labelled n has an energy n 2 {h 2 /SmL 2 ), The state of an
electron in a "hydrogen atom is fully determined if we give the
numerical values of the quantum numbers n, E, mg, s," 7 ^ n ' s
The principal quantum number and determines the energy
through £ = —/? H /n 2 {/? H being the Rydberg constant); H the
azimuthal quantum number, or the orbital angular momentum
quantum number, determines the magnitude of the orbital
angular momentum through the expression [2(fi + 1)] 'hand
the number of possible orientations of this momentum through
2£ + 1 ; m g selects which of the orientations the orbital angular
quantum numbers
191
momentum does in fact have {and therefore which of the
2fi + 1 atomic orbitals of given n and £ the electron occupies)
and is called the magnetic Quantum number (because if a
magnetic field is present the electron will have an energy
myj± & B on account of its orbital magnetic moment) ;s is the
"spin quantum number which determines the magnitude of the
spin angular momentum through [s{s + 1)1*6, and for an elec
tron s is fixed at the value s = j} m s , the magnetic spin quantum
number, determines the orientation of the spin angular
momentum relative to some axis, and if a magnetic field lies
along this axis there will be an additional contribution to the
energy of mj^S.
For atoms other than hydrogen the orbital energy also
depends on the value of £ on account of the effect of
penetration and shielding, but it is no longer possible to give a
simple analytical connexion between the values of the quantum
numbers and the energy of the state (but see quantum defect).
BOX 19: Quantum numbers
Capital letters are used for quantum numbers referring to
manyparticle systems.
F total angular momentum, including the contribution
of nuclear spin. Interpretation as for/.
/ nuclear spin quantum number, significance as for/. /
may have integral or halfintegral values, but one
isotope of an element has a single, characteristic value
of /.See Table 17.
I J total angular momentum (excluding nuclear spin), or
designation of a general angular momentum. J,J is
never negative, and may be integral or halfintegral
depending on the system,
magnitude of a.m.: hJ [/'(/' + 1)]
number of projections on a specified axis: 2/ + 1
magnitude of projection: see m.
If/ is composed from/j and/ 2 , then permitted
values are/ =/, + / a ,/, +/ 2  1, . . . /, / 2 .
K component of a.m. about symmetry axis of an
axially symmetric molecule: Kit K is restricted to
the 2/ + 1 values, J.J1, J.
S..L orbital a.m. quantum number (also called the
azimuthal quantum number}. Interpretation as for
/ but fi,L can take only integral values.
m,M magnetic quantum number (often appearing as nip,
m^ M L , M s , M f , Mj, etc.)
component of a.m. on a particular axis (by
convention the zaxis): mil,
energy of a magnetic moment in a magnetic
field B:mhyB.
n principal quantum number; the energy of an
electron in the hydrogen atom: —R In 2 with
n = 1, 2, . . .. (Note that £ cannot exceed n — 1.)
general quantum number in a variety of situations;
for example, the particle in a square well has
n = 1 , 2, . . . and energy n 2 V/8mZ. 2 ).
IV total angular momen turn, excluding electro n and
nuclear spin. Restriction and interpretation as for
U.
S,S °spin a.m. quantum number. Significance and
interpretation as for/V, and s,S may have positive
halfintegral values, s is a stnglevaluded intrinsic
property of a particle.
&$ spin projections for a spinj object, a corresponds to
m s  +~ and p 1 torn = \.
\A component of orbital a.m. about symmetry axis of a
linear molecule: Ah, All A,Aare restricted to fi, £—1,
. . ., — £ or L, L 1, ,..,—£. respectively. \— Xare
degenerate to a first approximation.
V vibrational quantum number. The energy of a
"harmonic oscillator is {JJ+ jlhoJo, where V ~ 0, 1,
2
0,2 component of spin a.m. about symmetry axis of a
linear molecule: oft, 2ft a can lie ats, s— 1, . . ., — s,
and 2 at S, 51 S.
lj,£2 component of total electronic a.m. about symmetry
axis of a linear molecule: ojfi, Slh. Q, can take the
2/+ 1 va\uesJ,J],...,J.
192
quantum theory: a synopsis
In Box 19 are listed some common quantum numbers and
the properties they determine: for further information, consult
the appropriate entry. The reader might care to reflect on why
quantum numbers are always either integral or halfintegral,
and newer fractions more vulgar nor numbers irrational.
quantum theory  , a synopsis. The view that energy could
be transferred between systems only in discrete amounts rather
than continuously arose from observations on the interaction
of matter and radiation and on the behaviour of solids at low
temperatures. The evidence came from the study of "black
body radiation, the "photoelectric effect, the Compton effect,
"atomic spectra (especially the spectrum of atomic "hydrogen),
and the "heat capacities of solids. The first quantum calculation
was due to Planck, who deduced the distribution law for
black body radiation. The first quantummechanical calculation,
where the quantum ideas were applied to a mechanical system,
was "Bohr's calculation of the energy levels of atomic hydrogen.
The early theory of quantum mechanics was displaced by a
new quantum theory in 1 926, when "Schrodinger proposed
his equation, and Helsenberg his "matrix mechanics. These
entailed a wholesale revision of classical physics, and gave a
theoretical basis to the waveparticle duality of matter. The
incorporation of relativity into the theory was made by
"Dirac, and relativistic quantum mechanics is now at the stage
of 'quantum electrodynamics and quantum field theory.
Further information .See MOM Chapter 1 for an outline of the
observations that led to quantum theory. See also Heisenberg
(1930) and Jammer (1966) for a historical perspective. An
interesting introduction has been given by Andrade e Silva
and Lochak (1969), A collection of the significant early
papers (in translation) has been prepared by van der Waerden
(1967).
quenching. The angular momentum of a system is quenched
when it is eliminated by the presence of some electrostatic
potential. In an atom, and for simplicity we fix our attention
on a hydrogen atom, the energy of an electron is independent
of the angular coordinates (its latitude and longitude), and so
its angular motion can occur smoothly and without hindrance.
In such a case the "orbital angular momentum remains constant,
and is well defined. When the atom is surrounded by ligands
the energy of the electron depends on its angular coordinates
and it experiences a force that accelerates it in a complicated
manner. In classical terms the acceleration continuously changes
the direction of the electron's motion and the average angular
momentum is zero.
One cannot use quite the same argument in quantum
mechanics because the trajectory is an alien concept, but one
can come to the same conclusion by considering the effect
of the ligand potential on the wavef unction of the electron.
The presence of the "ligand potential causes the electron to
collect in pools of high probability, either close to the ligands
if the potential is attractive or between them, if it is repulsive;
but the formation of these pools implies that the original
running wave has been turned into a standing wave {the
stationary antinodes being the pools), and with standing
waves there is associated no angular momentum. Therefore
the momentum has been quenched by the anisotropic
potential.
Questions. 1 . What does 'quenching of angular momentum'
mean? When does it occur? What causes it? What is its
quantummechanical explanation? Can the same explanation
account for the fact that in diatomic molecules only the
angular momentum about the internuclear axis is well
defined?
2. A wave running around thezaxis is described by the func
tion exp i/n0; the ^component of the angular momentum is
found by calculating the "expectation value of the operator
{h/i)d/30. Show that the zcomponent of the angular momen
tum for this state is mil. Now quench the momentum by
replacing the running wave, which has an even distribution of
probability, by the standing wave cos m<ji, which has accumu
lations of probability in the vicinity of = and f. What is
the expectation value of the zcomponent of angular momen
tum for this wave?
3. Using the hermiticity of the operator % g , prove that its
•expectation value is necessarily zero for real states. This is a
formal demonstration that angular momentum is quenched in
states represented by real wavef unctions.
Further information . Angularmomentum quenching is
described in more detail in MQM Chapter 6 and Question 3
is answered on p.417. Quenching profoundly affects the
magnetic properties of materials, in particular those of
transitionmetal ions, for when the orbital motion is eliminated
quenching 193
the paramagnetism can be ascribed solely to the spin magnetic
moments. These aspects of quenching are also described in
Earnshaw (1968), Ballhausen (1962), Figgis (1966), Jorgensen
(1971), and Griffith (1964). See also Davies (1967), and
van Vleck (1932).
R
radial distribution function. The radial distribution function
(r.d.f.) determines the probability that a particle will be found
somewhere within a spherical shell of thickness dr at the radius
r. For a wavefunction depending on only the radius the radial
distribution function Pir) is 4m 2 ^*{r)^{r), and the probability
of being in the shell of radius r and thickness dr is P(/)dr, The
source of this function can be traced by recalling the inter
pretation of the wavefunction and considering the response
of a probe sensitive to the presence of the particle. The prob
ability of finding a particle in the volume element dr surround
ing the point r is equal to ^*{r)^(r)dr, and a probe of volume
dr gives a response proportional to i^*(r)i^{r). Now let the
sensitive part of the probe consist of a thin spherical shell of
thickness dr and radius r. This probe is dipped into the atom
so that the nucleus is at the origin of the shell, and the meter
reading is indicative of the total probability of finding the
particle anywhere on the shell. The volume of the shell is
4ro 2 dr, and so if the wavefunction is isotropic, the reading is
proportional to 47rr 2 i/'*(r)i/'(r)dr, or P(r)dr. This means that
P(r) tells us the probability of finding the particle anywhere on
the shell of thickness dr at r.
The probe behaves in an odd manner, because its sensitive
volume diminishes as tt samples regions closer to the nucleus
(the volume of the shell is proportional tor 2 ). The meter
reading falls to zero at the nucleus, because there the shell
becomes vanishingly small, and it also falls to zero at very great
distances, because there the wavefunction vanishes. The
decaying exponential wavefunction of the hydrogen atom
therefore gives rise to a r.d.f. that rises from zero at the nucleus,
passes through a maximum, and then falls to zero at infinity.
This curve (see Fig. R1) shows the probability of finding the
electron at a particular radius irrespective of the angular
coordinates of the point: the square of the wavefunction itself
gives the probability of finding the electron at a specified
point, and the number of these points at a given radius increases
as 4w 2 .
Questions. What is the significance of the radial distribution
function (r.d.f.)? What is the difference in interpretation of
the r.d.f. and the square of the wavefunction? Discuss the
form and significance of the r.d.f. for the electron in the
1sand2sorbital5of the 'hydrogen atom. Using the math
ematical form of these two functions (Table 15) plot the
Radius
FIG. R1. The radialdistnbulian function for the ground state of
hydrogen.
194
corresponding r.d.f.. Deduce an expression for the most
probable radius of the charge distribution and discuss the
nature of the r.d.f. for anisotropic orbitals (for example,
the 2porbitals of hydrogen}.
Further information, A discussion of the r.d.f. will be found
in books that deal with atomic structure: see Chapter 2 of
Coulson (1961), §1.4 of Herzberg (1944), §4.10 of White
(1934), and §V.2 of Condon and Shortley (1963). Radial
distribution functions enable one to think sensibly about a
lot of inorganic chemistry, because they are the basis of
•penetration and shielding and consequent discussions. For
the trail through this matter 6ee Puddephatt's The periodic
table of the elements (OCS 3) , pp. 34, 35 of Pass's tons in
solution 3 (OCS 7), Earnshaw and Harrington's The
chemistry of the transition elements {OCS 13), and Phillips and
Williams (1965), particularly Chapters 1 and 2. Analytical
expressions and references to the r.d.f. of numerous atoms
will be found in McGlynn, Vanquickenborne, Kinoshita, and
Carroll (1972), especially Appendix B. See also Herman and
Skillman (1963) for numerical tables.
Raman Spectra. The Raman process is the inelastic scattering
of light by molecules. An inelastic process is one in which
energy is transferred between the two colliding systems. In
Raman scattering the light may deposit energy in the molecule
by exciting one or more of its internal modes (of rotation or
vibration), or it may collect energy from the molecules if a
mode is already excited. Since the internal modes are
"quantized the energy transfer is limited to welldefined
amounts, and so the scattered light contains frequency
components that are shifted from the incident frequency by
discrete amounts. The detection and recording of the frequency
composition of the scattered light constitutes the Raman
spectrum of the species.
In practice a brilliant, monochromatic beam irradiates the
sample and the light scattered in the perpendicular direction is
analysed: the forward scattered component would be
obliterated by the intense incident beam (but see below). The
spectrum consists of a strong component at the incident
frequency, the Rayieigh scattered component, which represents
the elastic collision of the light with the sample, and a series of
5branch
"I
Raman spectra
6B« — «
195
5
Stokes lines
r
2
Rayieigh
lire
Ob ranch
S 10
antiStates lines
incident frequency
FIG. R2. Rotational Raman spectrum of a linear molecule.
lines to high and low frequency (Fig. R2). The lowfrequency
lines are the Stokes lines, and arise from inelastic collisions in
which energy is transferred from the light to the molecule. The
lines on the highfrequency side of the Rayieigh component are
the antiStokes lines, and arise from inelastic collisions in which
molecular excitation is transferred from the molecule to the
light. The intensity of the Stokes lines is greater than the
intensity of the antiStokes lines because the latter depend on
the presence of molecules already in higherenergy states. It
should be clear that the spacing of the Raman lines contains
information about the vibrational or rotational levels of a
molecule, but to determine the information we must first decide
the significance of the transitions.
The Raman effect depends on the properties of the "polariz
ability of the molecule. This can be understood when the
scattering process is pictured in terms of the incident radiation
inducing a "dipole moment in the molecule, and this dipole
moment radiating electromagnetic radiation. The efficiency of
the process depends on the ease with which the molecule can
be distorted by the incident light, and hence It depends on its
polarizability. It can be understood on this picture that the
emitted light will not necessarily carry away from the molecule
all the energy needed to excite it: discrete amounts of energy
can stick to the structure. The molecule, then, must be
polarizable; but that alone is insufficient.
If the molecule is to show a rotational Raman spectrum
(where the inelasticity of the collision excites or deactivates
rotational motion) its polarizability must depend on its orien
196
Raman spectra
tation, A rotating hydrogen molecule is Raman active because
it has different polarizabilities parallel and perpendicular to
the bond. The methane molecule is rotationally Raman inactive
because, being spherical, its potarizability is independent of its
orientation and the field cannot couple to the nuclear frame
work. In the case of the elastic, Rayleigh scattering, the
requirement is much less stringent: all a molecule need be is
polarizable; therefore all molecules scatter elastically, but onl.
molecules with anisotropic polarizabilities can exchange
rotational energy with the light.
If the molecule is to show a vibrational Raman spectrum
the polarizability must change as the molecule vibrates. A
vibrating hydrogen molecule is Raman active because its
polarizability depends on how greatly the bond is stretched;
the antisymmetrical vibration of CO2 , which we can denote
CH«C— *0 ^ CH — C*«0, does not affect the polarizability of
the molecule (it leaves it roughly the same size) and so this
particular vibration is Raman inactive. As in the rotational
case, we see that the change in the polarizability is the
essential feature if energy is to be exchanged and the collision
be inelastic.
An important rule, the exclusion rule, applies to the
vibrational Raman spectrum of molecules having a centre of
symmetry. If in such a molecule there is a mode of vibration
active in the ordinary {infrared) "vibrational spectrum, then
that mode is inactive in the vibrational Raman spectrum;
conversely, if the mode is infraredinactive, it is Ramanactive.
It follows that the Raman effect is useful in the study of
vibrations (and rotations) that are inaccessible to normal
absorption spectroscopy.
The scattered radiation of the Raman effect is polarized
even if the incident radiation is not. Fig. R3 illustrates the
simpler case of Rayleigh scattering, which is the elastic
scattering of light: the Raman scattering process is analogous,
but slightly more involved. From Fig. R3 a it should he clear
that if the molecule is isotropic and the incident light plane
polarized the scattered light is also planepolarized. The
scattered light is also planepolarized if the incident light
is unpolarized. If the molecule is anisotropic (Fig, R3 b),
the scattered light has both polarization components, and
so is not planepolarized. A convenient measure of the degree
of polarization is the depolarization ratio p, which is defined
FIG. R3. Depolarization of Rayleigh lines and, by analogy, of Raman
lines.
as the ratio of the intensities / ( and ^as defined in Fig. R3.
In the isotropic molecule case there is zero intensity in the
scattering plane, and so p = 0. In the anisotropic case there
is intensity both in the plane and perpendicular to it, and
the depolarization ratio is nonzero. For a freely rotating
molecule the maximum value is p = 7 for unpolarized
incident light, and p=  for planepolarized incident light
{for the polarization and geometry of Fig. R3). The
polarization in Raman scattering is determined similarly,
but it is necessary to consider the isotropy or anisotropy of
the changes in the polarizability of the molecule. Thus a
completely symmetrical vibration plays in the Raman case the
role of the symmetric molecule in the Rayleigh case; therefore
the Raman scattering from such a mode is fully polarized and
p = 0. If a vibration is anisotropic the Raman scattered light
is depolarized, and so p > 0. If the incident light is polarized
p cannot exceed , and if it is unpolarized p cannot exceed
7. It follows that the determination of the polarization of the
Raman scattered light is a valuable tool for determining the
symmetry of the active molecular vibrations.
We still do not know what transitions the Raman lines
represent apart from the qualitative remark that they represent
modes involving a changing polarizability. The 'selection rules
for Ramanactive transitions are that the "vibrational quantum
number of a mode must change by + 1; the Totationat quantum
number must change by AJ = + 2, The twoquantum rotational
Jump behaviour stems from the fact that the transitions depend on
the polarizability rather than the 'transition dipole moment
(which determines the absorption spectrum). One approach
to understanding the occurrence of 2 is to picture the scattering
process as involving two dipole transitions, one for the photon
coming in and one for the photon going out. A classical picture
elucidates why the rotational quantum number changes by
± 2 but the vibrational quantum number changes by ± 1 : in a
rotation of a molecule the dipoie moment is restored to an
indistinguishable orientation after a complete rotation of the
molecule, but the polarizability is restored to an indistinguish
able orientation twice on a revolution. In a vibration the dipole
moment and polarizability return at the same time and
vibrational Raman and absorption spectra both have Av = ± 1.
The rotational Stokes and antiStokes lines are related
to the "branches of a rotationvibration spectrum; and anal
ogously to the notation used there, they are referred to as the
Obranch (AJ = 2; antiStokes) and the Sbranch {AJ m +2;
Stokes). Knowing what transitions the spectrum shows it is
possible to relate the Raman lines to the energy levels of the
molecule by the same analysis as in the corresponding absorption
spectrum, and so to extract "force constants, moments of
inertia, and molecular geometry, and to identify unknown
species.
An important experimental requirement in Raman
spectroscopy is an intense monochromatic light beam; what
is more natural than to apply lasers, which have just this
property? LaserRaman spectroscopy is today a major branch
of study; not only does it refine the conventional Raman
technique by enabling frequencies very close to the exactly
defined incident frequency to be studied, but its unique
properties give rise to a number of new effects. The properties
put to work are the low divergence of the beam, which enables
observations to be made close to the forward direction, and
its exceedingly high intensity.
In the stimulated Raman effect an intense laser beam is
focused on a sample, and the light scattered in the forward
direction, or just off axis, is observed. The light in the forward
Raman spectra
197
direction itself is the same as the incident frequency, but
mixed with it is Stokes radiation of frequencies v,v— v.,
V— 2l> { etc., where v, is the frequency of an internal mode
giving an intense Raman signal in a conventional experiment.
Surrounding this narrow, forward scattered beam are a series
of concentric circles of light of increasing frequency; the
first ring is of frequency v + v the next ring has frequency
v+ IV y the third P+3i>, and so on (Fig. R4 a). The effect
arises from the fact that the initial Stokes scattered line is so
intense that it can be scattered again, and an inelastic collision
leads to a further Stokes line at V — 2v.; and so the process
continues as more quanta are chipped off the beam. Likewise
the intensity of the antiStokes lines is due to successive scat
tering, and the initial high intensity of the antiStokes light at
u+y. f is due to the high intensity of the initial beam permitting
the annihilation of two V photons by one molecule, followed
by the simultaneous generation of one of frequency V + V. and
another of frequency V — V y Only for very high beam intensities
is there a significant probability that two photons can simul
taneously be in the region of the same molecule, which is
necessary if the frequencysharing is to occur. The angular
FIG. R4. (aj Stimulated Raman effect, (b! HyperRaman effect.
Ic) Inverse Raman effect.
198
Ramsauer effect
dependence of the frequency of the scattered light is a conse
quence of the conservation of momentum in the collision.
In the hyperRaman effect an intense beam of frequency P
is focused on a sample, and together with the normal Raman
scattering there appear frequencies of 2i>and 2v± t>, (Fig. R4 b).
Thus the hyperRaman effect is the inelastic scattering struc
ture on highfrequency photons generated by the annihilation
of two lowfrequency photons in a simultaneous event involv
ing one molecule and two incident photons. As in the stimu
lated Raman effect, the efficiency of the hyperRaman effect
depends on the intensity of the light. One application is to the
measurement of molecular hyperpolarizabitities (see
"polarizability).
In the inverse Reman effect two beams of light are employed
and focused on a sample. One has a continuous spectrum and the
other is a highly intense monochromatic beam. It is found that
transitions occur which appear as absorptions from the
continuum at frequencies V + v . The process may be pictured
as the arrival at the molecule of a photon of the continuum
with a frequency V+ V; simultaneously there arrives a laser
photon of frequency V which stimulates the excited molecule to
shake off a photon of its own frequency, leaving behind an
amount of energy corresponding to the frequency V. (Fig. R4 c).
Questions. 1 . What is the difference between an elastic and an
inelastic process? On which does the Raman effect rely? What
experimental arrangement is employed in Raman spectroscopy?
In what way can the use of a laser benefit the observations?
What is meant by the Rayleigh component? Discuss the
appearance of the Raman spectrum and indicate the signifi
cance of the Stokes and anti Stokes lines. On what molecular
property do the Raman and Rayleigh scattering rely? Why do
they so depend? What is a necessary property of the molecule if
it is to show a vibrational Raman spectrum, and a rotational
Raman spectrum? State the exclusion rule. Which vibrations of
carbon dioxide are Raman active, and which are infraredactive?
State the selection rules for vibrational and rotational Raman
spectra: why does the latter reiy on a doublequantum jump,
but the former on a single jump? What information can be
obtained from the splittings in the 0 and Sbranches of a
rotational Raman spectrum?
2. Show that in a classical polarizable rotator the induced
dipole moment will emit Stokes and anti Stokes lines. In the
calculation suppose that the polarizability of the molecule
varies harmonically between a — 6a and a + 5a about a mean
value a. Proceed by showing that the molecule possesses an
induced dipole moment of the form aE + oofcostor, where
E is the imposed optical field; then take the timedependence
of £ to be costdof, and show by simple trigonometric manipu
lations that the overall induced dipole has components,
oscillating at OJ, oj± av This calculation leads to the prediction
of equal intensities for the Stokes and antiStokes lines: why
is that false?
3. "Grouptheoretical arguments may be applied to the Raman
problem in order to determine the vibrational selection rules.
The transition operator is the polarizability tensor for the
molecule: this transforms likexx, yy, and zz. Show, using
Box 4 on p. 33, that in a centrosymmetric molecule the exclusion
rule follows from the difference in symmetry of the polariz
ability and the dipole moment under inversion. From the
characters in Table 3 confirm that the A i( E, and T 2 vibrations
are Raman active in a tetrahedral AB 4 molecule. What trans
itions are Raman active in water, carbon dioxide, and
ammonia?
Further information. See MQM Chapter 10 for answers to
Questions 2 and 3 and for further discussion. For applications
see Woodward (1972), Wheatley (1968), Whiff en (1972),
and Barrow (1962). LaserRaman spectroscopy is described in
Long (1971) and Gilson and Hendra (1970). See Herzberg
1 1945) for further details. Woodward's Chapter 19 is a good
summary of depolarization processes.
Ramsauer effect. When a beam of electrons was passed
through a sample of argon and the other noble gases, it was
found that the scattering power of the sample decreased
strongly at some energies of the electron beam (a transmission
resonance is observed at about 0.7 eV). This effect may be
understood in terms of the 'wave nature of the electron, and
its decreasing wavelength as its energy increases { °de Broglie
relation). The system may be viewed as an electron wave
incident on atoms, and these are regions of potential different
from the surrounding vacuum. But in regions of different
potential energy the wavelength of the electron is changed
refractive index
199
(see kinetic energy). Just as in the case of an incident beam
of light passing into a region where the refractive index
changes, there are reflections from the front of the atom
(where the potential changes abruptly), and reflections from
the opposite inside surface where the potential drops back to
the vacuum level. If the potential has a thickness equal to
one quarter of the wavelength of the electrons the waves
reflected backwards from the two surfaces interfere destruc
tively, but the waves transmitted interfere constructively. It
follows that the intensity of the electrons reflected by the
atom is reduced and that the transmitted intensity is increased.
(This of course is the role of a coating on a lens.)
Further information . A helpful discussion of transmission
resonances is given in §11. 8 of Bohm {1951} and applied to
the Ramsauer effect in §11. 9 and §21.54, where he explains
that the effect is not quite the same as the square well process
we have described. See also §99 of Davydov (1965) and Mott
and Massey (1965).
refractive index. The ratio of the speed of light in vacuo c to
its speed in a medium v is the refractive index of the medium:
n = civ.
The size of the refractive index depends on the strength of
the interaction between the light field and the medium, and as
the electric field of the light has a stronger interaction than
the magnetic field, we should expect the refractive index to
depend on the electric "polarizability of the medium. The
greater the polarizability, the stronger the interaction, and the
greater the drag on the progress of the light. This guess is
confirmed by calculation, for in a nonpolar medium the
refractive index is related to the relative permittivity (dielectric
constant) by ri 1 = € r , and the relative permittivity Is related
to the polarizability by e r = 1 + Nct(Uile , where N is the
concentration of molecules and a(co)their polarizability. But
we have to be just a little careful because the molecule on
whose polarizability we might at one instant focus our
attention is surrounded by other polarizable molecules. These
other molecules respond to the electric field of the light and
their polarization enhances the field experienced by the
central molecule. Therefore we should apply a correction to
take into account the presence of the surrounding molecules.
This is the Lorentz local field correction, and it involves
increasing the strength of the field by a factor of  (e +2),
e is the relative permittivity, or n , and so the expression
for the refractive index becomes rather more complicated.
By an unlikely but helpful coincidence the indistinguishable
Lorenz and Lorentz introduced this correction independently
and simultaneously, and arrived at the Lorenz Lorentz formula
(n 2  1 )l{n 2 + 2)p = Afa:M/3pe o , where p is the density.
Since N is proportional to the density the righthand side of
the equation is independent of the density, and so too
therefore is the term on the left, which is called the refractivity
of the medium.
It should be noticed that the refractive index (and the
refractivity) depends on the frequency of the light through
the dependence of the polarizability on the frequency. This
dependence is described in the section on "polarizability, but
at optical frequencies is due to the highenergy photons being
more able than lowenergy photons to excite the molecules
into their lowlying excited electronic states. Therefore, as
the frequency gets greater (and approaches an absorption
frequency of the molecule), the interaction gets stronger and
the refractive index gets larger. For this reason the refractive
index for blue light exceeds that for red light, and, as a
consequence of this, a beam of white light is dispersed by a
refracting medium. The name dispersion, which denotes the
frequencydependence of a property, is derived from this
aspect of the refractive index.
Very close to absorption bands the refractive index varies
strongly. If the behaviour of the refractive index throughout
the frequency range is known it is possible to extract the
absorption spectrum of the molecule and vice versa. The
formula that enables this to be done is the Kramers Kronig
dispersion relation. A dispersion relation relates the overall
frequencydependence of a dispersion property (such as
refractive index) to the absorption property and vice versa.
The refractive index for a composite molecule is the sum of
the refractivities of its parts in so far as the polarizability of a
molecule is itself an additive property.
Further information. See MQM Chapter 1 1 for an account in
more detail of the derivation of the quantummechanical
expression for the refractive index of the molecule. See
200
Renner effect
Corson and Lorrain (1970) for a derivation via the Clausius
Mossotti equation of the Lorentz localfield correction. See
also van Vleck (1932) for a discussion. A simple introduction
to the KramersKrdnig dispersion relation will be found in
Slichter (1963), and tougher accounts in Roman (1965, 1969).
Pertinent information on refractive indices is contained in the
polarizabitity Box (Box 17 on p. 179),
Renner effect. The Renner effect, or as it is sometimes more
fairly called the RennerTelter effect, is an interaction between
the electronic and vibrational motions of a linear molecule
(especially a triatomic linear molecule) which removes the
"degeneracy of the energy levels. Consider a IIstate of a linear
triatomic; bending the molecule into a triangular conformation
affects the two components of the electronic molecular
orbital differently. For simplicity let the molecule have a
single electron in a jt orbital; in the linear case the it x and
it orbitals are degenerate, but when the molecule is bent
they diverge in energy: three possible types of behaviour are
illustrated in Fig. R5. The Renner effect appears in the
spectrum of the molecule because the bending vibrational
levels are modified by their interaction with the electronic
levels.
been given by Renner (1934), Pople and LonguetHiggins
(1958), and LonguetHiggins (1961).
resonance. The concept of resonance has its roots in classical
mechanics, and it is helpful to recall that application of the
term. If two pendulums are weakly linked (for example, if
they hang from the same slightly flexible axle, as in Fig. R6)
the motion of one is experienced by the other. If one is
initially still, and the other set in motion, the energy of the
latter will be transferred to the first, which will begin to
swing, and pass back to the second some of its acquired energy.
This ebb and flow of energy continues indefinitely in the
absence of damping forces. The exchange of energy is most
effective when the pendulums have the same natural frequency,
and this condition of equality is known as resonance; the
energy, or amplitude, is then said to resonate between the two
tuned systems.
It is possible to imagine a form of coupling where the energy
of the system is lower if the pendulums swing in phase, and
where the worst arrangement (in terms of energy) occurs when
they swing in opposition. A flexible axle is an example: it
might require less energy to twist the support in the same
direction at each pivot than to twist it in opposition (Fig R6);
O^X)
FIG. R5. Renner effed splitting
of irlevels: three possible
situations.
Further information . An account of the Renner effect in
molecules is given in §1.2 of Herzberg (1966), who provides
a number of examples, and gives expressions for the energies
of the vibronic states of the molecules showing the effect. See
also §10.13 of King (1964). A quantitative treatment has
if there is no coupling between the periodic systems their
relative phase is immaterial.
With the preceding classical picture of resonance and
coupled systems in mind we shall consider two examples of
quantummechanical resonance. The first is the interaction of
FIG. R6, Resonance of two coupled pendulums. The energy splitting A
is greatest when the natural frequencies are identical.
light with atoms (or molecules) . The atom and the electro
magnetic field (light) play the roles of the two coupled
periodic systems. The 'natural frequencies' of the atom are
the transition frequencies, and we can imagine adjusting the
'natural frequency' of the electromagnetic field bathing the
atom by changing the frequency of the incident light. There
comes a point when the frequency of the light matches the
frequency of a transition within the atom: the combined
system behaves like the two coupled pendulums, and energy
is transferred from the light field (resulting in absorption) or
to the light field if the atom is already in an excited state
(emission). These processes occur most effectively at resonance.
The other example of resonance is the type one encounters
tn theories of molecular structure, particularly in the "valence
bond theory, where one attempts to describe the true structure
of a molecule by a superposition of simple, canonical structures.
The bestknown example is benzene, where one attempts to
describe the structure by the superposition of the two Kekule
structures. Let us suppose that there is a coupling between the
Kekule forms; then the superposition will have a lower energy
than either form alone if the phase of coupling is correct
(remember the second aspect of the classical idea of resonance).
resonance energy 201
Thus each Kekule' structure behaves like a simple pendulum (by
symmetry they correspond to the same frequency) and the
presence of a coupling means that their energy is different in
conjunction than separately. The nature of the coupling may
be visualized as the tendency of the n electrons in one C— C
bond to push the neighbouring irelectrons into a neighbouring
gap in the jrbonding structure: in this way one Kekule structure
is turned into the other (Fig. R7) (see "benzene and Fig. B8).
Thus resonance stabilizes in the sense that the coupled system
has a lower energy; the stabilization is greatest when the natural
frequencies of the separate systems are the same (consequently
the resonance stabilization of benzene is large), loniccovalent
resonance also stabilizes, but as the contributing forms differ
in energy the resonance is less exact and the effect less.
our of phase
dec trostofic
coupling
in phase
FIG. R7. Resonance in VB theory (of benzene). E denotes ihe
resonance energy.
Further information. For discussions of the concept of resonance
in molecularstructure studies, see §5.5 of Coulson (1961), §1.3
of Pauling (1960), and §13 of Eyring, Walter, and Kimball
(1944). Resonance in light absorption is discussed in Chapter 7
of MQM, and very well illustrated by the phenomena of
"electron spin resonance and "nuclear magnetic resonance. A
transmission resonance is described under "Ramsauer effect.
See also "benzene, "valence bond, and "resonance energy.
resonance energy. An "aromatic molecule is more stable than
untutored speculation, confronted with a molecule bristling with
double bonds, might predict. The difference between the true
energy, which takes into account the stabilization of the
molecule by "resonance among various structural possibilities,
202
rotation of molecules
and some reference state of the molecule, is termed the
resonance energy (see Fig. R7). {An alternative name, which
reflects the analogous phenomenon in 'molecularorbital
theory, is derealization energy.)
The choice of reference is difficult and several suggestions
have been made. The Huckel definition is the most elementary:
it defines the resonance energy as the difference in energy
between the true molecule and the hypothetical molecule in
which there are localized ethenetype double bonds. Thus if
the jrelectron energy of 'benzene is found to be 3a + 40, and
the energy of each of the three ethene bonds is a + j3, the
resonance energy is 3.
The difficulty with the Hu'ckel definition is the arbitrariness
of the comparison: would it not be better to attempt to find
the energy of the unconjugated form of the molecule itself, and
then use this as the reference? Why not take the hypothetical
cyclohexatriene molecule, calculate its energy, and use it as
the reference for benzene? This reference molecule has
alternating short and long bonds, and it is argued that by its
use the full effect of conjugation, derealization, or resonance
on both the 7T and the aelectrons is taken into account. This
is the basis of the MultikenParr definition of resonance energy.
A common method of measuring the resonance energy is to
determine the heat of formation of the molecule (for example,
by determining the heat of combustion of the molecule in a
bomb calorimeter, and knowing the heats of formation of
the combustion products, which are often carbon dioxide
and water), and comparing this result with the value obtained
on the basis of ascribing a bond energy to each bond in the
structure (these bond energies may be found from tables): the
difference of the two calculations is the resonance energy. For
example, in benzene a Kekule structure has six C— H bonds, three
C— C bonds, and three C=C bonds; the energy of that structure
is therefore 6F(CH) + 3F(CC) + 3£(C=C). The difference
of this value from the observed heat of formation is the
resonance stabilization of the Kekule' structure (150 kj mof 1 )
Modern values are often obtained from the heat of hydrogen
ation, which is less in the presence of resonance stabilization
(the more the molecule is stabilized the closer it lies to the
fully hydrogenated energy). In this type of determination a
molecule containing three (unconjugated) double bonds is
expected to have a heat of hydrogenation three times that of
cyclohexene;the observed heat of hydrogenation of benzene
is less than that figure by 1 50 kJ mol 1 , and this difference is
identified as the resonance stabilization of the Kekule form.
Questions. What is meant by resonance energy? What are two
possible definitions? Can you think of alternative definitions?
How would you determine the resonance energy of a hydro
carbon? What changes in the value of the resonance energy
might you expect on replacing the Hu'ckel definition by the
MullikenParr definition? The heat of formation of the
naphthalene molecule was measured and found to be 8623 kJ
mol 1 . The following bond energies have been measured in
other experiments: CC, 333 kJ mol" 1 ; OC, 593 kJ mof 1 ;
CH, 418 kJ mof' . Calculate the resonance energy of
naphthalene. On the basis of the Hu'ckel molecularorbital
scheme, estimate the resonance energies of cyclobutadiene and
butadiene {buta1, 3diene).
Further information. See Streitweiser (1961), Salem (1966),
Dewar (1969), Murrell and Harget (1972), and McGlynn,
Vanquickenborne, Kinoshita, and Carroll (1972) for further
discussion. A short table of resonance energies will be found
in Chapter 9 of Streitweiser {1961 1 and Chapter 4 of Murrell
and Harget (1972). A book devoted to resonance in organic
chemistry is that by Wheland (1955.
rotation of molecules. The rotational energy of a molecule
arises from its 'angular momentum, and because the latter is
quantized, so too is the rotational energy. The energy separ
ation of adjacent quantized levels is small, and transitions
between them occur in the microwave region of the spectrum
(10 8  10 n Hz, 10 1  1CT 2 cm). The separations are deter
mined by the moments of inertia of the molecule, and so
microwave spectroscopy gives information about molecular
geometry. Rotational transitions are also observed in conjunc
tion with the "vibrational, 'electronic, and "Raman spectra of
molecules, and these give the same kind of information.
Molecules are normally classified into four groups In a dis
cussion of their rotational energy levels: linear molecules,
symmetric tops (molecules with an axis of symmetry),
spherical tops (spherical, tetrahedral, and octahedral molecules),
and asymmetric tops {anything else).
rotation of molecules
203
1. Linear molecules. A linear molecule has only two rotational
degrees of freedom: these correspond to endoverend rotation
about two perpendicular axes. It is easy to see why the question
of a third degree of rotational freedom does not arise: in a
diatomic molecule the atoms have six degrees of freedom;
three are ascribed to the overall translation of the molecule
one is ascribed to the vibration of the bond; and so only two
remain for the rotations. A similar argument applies to any
linear molecule. The classical kinetic energy of rotation of a
body of moment of inertia / is jlco 2 , and as the classical angular
momentum is /tothis energy may be expressed as {/Gj} 2 /2/.
The transition to quantum mechanics is now trivial, for the
quantum theory of angular momentum tells us that the
angular momentum of a body is limited to the values
[J{J + 1 )] '^h with J confined to the integers 0, 1 , 2 There
fore it is merely necessary to replace /toby this expression to
obtain the energy of the Jth quantized rotational level as
jy + 1 )h 2 /2/. The quantity h 2 /2/ is normally written B and
called the rotational constant oi the molecule. It is important to
note that the separation of the rotational levels decreases as
the moment of inertia of the molecule increases (for B then
decreases} and that the level J lies at an energy 2BJ beneath
Its neighbour. Typical values of B are 60.809 cm" 1 for Hj,
30.429 cm" 1 for D 2 , 10.5909 cm" 1 for HCI, and 0.0374 cm" 1
for l 2 ; more values are given in Table 10.
In accord with the theory of angular momentum there are
2J + 1 possibilities for the orientation of the rotational angular
momentum vector of the molecule, but in the absence of
external fields the orientation of the molecule has no effect on
its energy, and so 2J + 1 orientations all have the same energy
(they are "degenerate).
Pure rotational transitions can be stimulated by an electro
magnetic field of the appropriate frequency only if the
molecule has a permanent dipole moment, for the dipole acts
as a kind of lever for the interaction, and through it the field
accelerates the molecule by exerting a torque. Molecules
without dipole moments (including, for example, H 2 and C0 2 )
do not show a pure rotational spectrum. If a molecule has a
permanent dipole the field can induce transitions only between
neighbouring levels; that is, the "selection rule for rotational
transitions is &/ = +1 (absorption) or A./ = — 1 (emission). The
rotational spectrum is therefore a set of fines spaced by 2B,
J: 6
J ^=L
1 2 3 4 5 6
FIG. R8. Pure rotation spectrum of a linear molecule, showing
population of levels and the intensity distribution (which is determined
by the population and the size ol transition dipole moments).
with an intensity distribution governed by their initial (thermal)
population and, in a slightly complicated way, by selection
rules which vary with J (Fig. R8).
2. Symmetric tops. Symmetric tops may be either prolate
[cigar shaped) or oblate (disc shaped), and have three rotational
degrees of freedom. As they have two distinct moments of
inertia (one parallel to the figure axis /, , , and two equivalent
moments perpendicular to the axis L, the energy of rotation of
the molecule depends on how the angular momentum is
distributed about the three molecular axes. If the molecule
rotates endoverend the energy is determined by /j_alone, but
as the motion becomes more like the spinning of a top about
its axis, so the energy becomes dependent more strongly on I...
The amount of rotation about the figure axis is determined by
a quantum number K: Kh is the component of the angular
204
rotation of molecules
FIG. R9. Rotation of a (prolate) symmetric top. When K is large the
motion is largely about the figure axis; when K is small it is largely
about the perpend icu tar axis.
momentum on the axis, and can vary in integral steps from
+J to —J (there are 2 J + 1 such values) (see Fig. R9). When
K = there is no rotation about the axis. It is quite easy to
see how the quantum number K enters the problem: the
classical energy of rotation depends on the angular momentum
3bout each of the axes, If with the axis q there is associated a
component of angular momentum Uoj) and a moment of
inertia / then the kinetic energy associated with this mode is
~{taj) 2 ff . Therefore the total kineticenergy of our molecule
is J (/to}^// L + \ V<4 2 Jlj_+ \ U<4j// n , where z is taken to
be the figure axis. If we recognize that \I<A\ + (/to)^ + {tco) 2 ,
is the square of the total angular momentum of the molecule,
which can be identified with the quantummechanical value
J{j + 1)fi 2 , it is possible to cast the energy expression into the
form JU + \\B + j(1//„  1//J {100)1, where B is h2/2/ _i_
Next we identify (l(+>) z as the component of the angular
momentum on the figure axis, and from our knowledge of
the quantum theory of angular momentum recall that all
components of angular momentum are quantized, and so set
this component equal to some integral multiple of h: thus we
identify {/w) r with KTi.
The final form of the energy expression is given in Box 20.
The form of the expression is as we anticipated, for when K
is zero the energy is determined solely by (j_, and when K  ±J
the energy is determined largely (but not solely) by r"„. It is
also satisfying to note that the energy depends on K 2 rather
than K itself: this means, as common sense requires, that the
direction of motion about the figure axis is immaterial. Remem
ber that the molecule can still have 2J + 1 different orien
tations with respect to some spacefixed axis, but these are
degenerate in the absence of external fields.
The spectra of symmetric tops depend on the presence of
a permanent dipole to act as a lever: by symmetry this dipole
must lie along the axis and so a light wave is unable to exert
a torque to accelerate the motion about the axis; consequently
the "selection rules are AK = and AJ = ±1.
3. Spherical tops. The energy levels of spherical tops can be
obtained very easily from the expression for the symmetric
top because we may set /j_= /,, . The resulting expression is
given in Box 20. The energy is independent of K because all
axes of the molecule are equivalent. Another way of saying
this is that all the 2J + 1 states K for a given value of J are
■degenerate. Therefore the rotational levels of the spherical
top are [2J + 1 ) 2 fold degenerate (because the 2J + 1 space
orientations, labelled M, are also degenerate). A spherical
top is invisible in microwave spectroscopy because it has no
permanent dipole, and so cannot be accelerated by incident
radiation. The rotational levels of this and the other types
of molecule can, of course, be excited by collisions with
other molecules, or the walls of a vessel, and so the large
number of accessible rotational levels must be taken into
account when the properties of the gas (such as its heat
capacity) are calculated.
4, Asymmetric tops. These are real horrors because the angular
momentum cannot be distributed among the three axes in a
tidy fashion. If we think of a prolate symmetric top with a
bump added to one side we shall see the problem. If the top is
spinning about its figure axis the effect of the bump will be to
tip the molecule away from its original direction; that is, the
bump induces the transfer of angular momentum from the
figure axis to the other two axes. But the process continues.
rotation of molecules
205
BOX 20; Rotational energy levels
Diatomic (and linear} molecules
E{J)=BJ(J+%) B = tf/2t.
(B is in joules if / is in kg m 2 ; B = h/400ffi:/ is the
appropriate expression in cm" 1 .)
Transition energy: A£(J) = E (J + 1)  E{J) = 2SI/ + 1 ),
Symmetric tops
EU, K\ = BJU + 1) + (A  B)K 2
B = h 2 /2/ ± A=h 1 /2I U .
Transition energy: Af (J) = £(J + \,K)— EU, K)
= 2B{J + 1).
Spherical tops
E(J) = BJ{J+\) B = \\ 2 I2I.
Moments of inertia:
diatomic molecule: {m ,m B separated by R)
J m A m B \ ,
linear triatornic molecule: (m m , m separated by
fl A8 and V
l = m tf +m R 2 to^Mm^R ) 2
m A rt AB^ m c W BC (BI A +/» a +m c )
pyramidal molecule: (AB 3 , bond length R, BAB
angle 0)
t n = 2m B R 2 i\co$0)
« m .m B R 2 (1 + 2 cos&)
J B (3m. +m D )
and although the total angular momentum remains constant,
the molecule rotates in a complicated, varying pattern. The
energy levels can be obtained only by complicated techniques,
and the rotational spectra are immensely complex. Neverthe
less, information about molecular geometry can be extracted
by a close analysis of the shapes of the spectral bands.
Questions. 1 . What is the source of the rotational energy of a
molecule? Why is it quantized? Is there a zeropoint energy? In
what region of the spectrum do rotational transitions occur?
How do the transition energies depend on the size of the
molecule? What information can be obtained from rotational
spectra? What must a molecule possess if it is to absorb
radiation by a rotational transition? What is a classification of
molecular types? What is the significance of the quantum
number K? When does the rotational energy of a molecule
depend on the space projection Ml How manyfold
degenerate are the rotational states of linear molecules, and the
various kinds of tops? Why does the energy of a symmetric top
still depend weakly on /^when K = ±J? Can the rotational
energy of a symmetric top ever be ascribed solely to motion
about its figure axis? What are the selection rules for rotational
transitions? How do all the rotational modes come into thermal
equilibrium even though some transitions are electricdipole
forbidden?
2. Calculate the energylevel separation (in cm 1 ) for the ro
tational levels of H 2 , HD, and D 2 . taking the bond length to be
0074 nm in each case. The pure rotational spectrum of HI
consists of a series of equally spaced lines with a separation
of 12.8 cm 1 ; calculate the bond length of the molecule. Why
are the lines equally spaced? {Draw an energy level scheme
and apply the appropriate "selection rules; or just think about
2BJ and Ay.)
3. Calculate the relative populations of the rotational levels
of H 2 ,D 2 , l 2 (5 = 0.0374cm" l ),CH<, (r CH =0.1094 nm),
and NH 3 (/,, = 4437 X 10 47 kg m 2 , l ± = 281 6 X lO" 47
kg m 2 ) at 300 K and at 1000 K. Use the Boltzmann distri
bution, include the degeneracy of the levels properly, and
forget (if you already know) about "nuclear statistics.
Further in formation . A fuller discussion of molecular rotation
and rotational spectra is given in MQM Chapter 10. See also
Whiffen (1972), Barrow (1962), and King (1964). An
introduction to microwave spectroscopy has been written by
Sugden and Kenney (1965), and a standard work is that by
Townes and Schawlow (1955); both books give a bibliography
of molecules studied. Allen and Cross (1963) give a theoretical
discussion. See Herzberg (1945, 1950, 1966) for further
discussion and applications; in these volumes are useful collec
206
Rydberg constant
tions of molecular data (such as bond lengths, "force constants,
and rotational constants). A useful collection of molecular
structural data has been compiled by Sutton (1958). Rotational
transitions can also be studied by observing the "vibrational and
"electronic spectra of molecules, and by thG use of "Raman
spectra. The "heat capacity of molecules depends upon the
accessibility of rotational states, and therefore on their energy.
A complicating feature of rotational problems is that the
occupation of rotational states is restricted by the "Pauli
principle: for more information see "nuclear statistics.
Rydberg constant. The Rydberg constant relates the energy
of an electron in a hydrogen atom to its principal "quantum
numbers: E = —Rlri 1 . It is necessary to be just a little careful
in recording values of R because it depends on the mass of the
electron and the proton, and if some oneelectron atom other
than hydrogen itself is being considered it is necessary to
correct it for the mass of the new nucleus. That having been
BOX 21:
Rydberg constant
Rydberg constant
rV = &m a e*c 3 m 3
R' m C
1097 373 X 10 7 m" 1
1097 373 X 10 s cm"'
3289 842 X 10 1S Hz
fl „ = R l hc = 2™ e c2a
'■ 2179 72 X 10~ 18 J
1313kJmol"'
1360 eV,
Hydrogenatom Rydberg constant
r' = R'l(\ + m Im ) 1096 776 X 10 7 m" 1
1096 776 X 10 s cm *.
The value of fl_ is referred to as a rydberg {1 Ry ~ 13 60 eV)
Note that 1 Ry is half the atomic unit of energy (the hartree
E : IE =2Ry).
a a
said, we refer to the true Rydberg constant as the quantity
R in Box 21 . The Rydberg constant for the hydrogen atom,
taking into account the finite mass of the proton, is
ft = r /(i + m Im ); this is also recorded in the Box. The
H w a P
Rydberg constants for other nuclei can be obtained by
replacing the mass of the proton in this expression by the
mass of the nucleus of interest. These different expressions
arise because the electronproton system rotates around its
centre of mass, which is slightly shifted away from the position
of the nucleus by virtue of its finite mass and the electron's
nonzero mass.
Further information. See Bohr atom for the first calculation
of the Rydberg constant in terms of fundamental constants,
and "hydrogen atom for the basis of its quantummechanical
deduction. Both entries give further information.
Rydberg level. An electronic transition in a molecule might
lift the electron out of the valenceshell orbitals into an outer
orbital: the state so formed is a Rydberg state, and the electron
occupies a Rydberg level. An example of this would be the
excitation of a 2pelectron of the fluorine atom in the fluorine
molecule into a 3sorbital, or something higher.
The Rydberg levels are of interest in so far as the important
electron is in a very diffuse orbital; so diffuse, in fact, that in
a diatomic molecule the two nuclei appear to the electron as
a single nucleus. This implies that the Rydberg electron is
only very weakly coupled to the nuclear framework, which is
therefore able to rotate without dragging the electron round
with it: see "Hund's coupling case (d). Rydberg electrons are
characterized by small quantum defects: since they are so
diffuse they hardly interact with the inner electrons and their
wavefunctions resemble those of the "hydrogen atom.
Further information . Rydberg levels and states are discussed in
some detail in §10.3 of King (1964) and gVI.Bof Herzberg
(1950). For a thorough discussion see Duncan (1971).
s
Schrodinger equation. The Schrddinger equation, which by
one of those rare coincidences is named after him who did
indeed discover itt, is a differential equation whose solution
is the "wavefunction for the system under consideration. This
implies that it is of central importance, for once we have the
wavefunction all the properties of the system are, in principle,
predictable, because the structure of quantum mechanics tells
us how to elicit the Information. The application of quantum
mechanics to physical systems therefore boils down to solving
the appropriate Schrddinger equation, and realizing that the
mathematical function which is the solution is the wave
function for the system.
Unfortunately the Schrodinger equation is not a simple
algebraic equation [like x 2 = 2), but, as mentioned above, it is
a differential equation. Except in a fairly small number of cases
such equations are very difficult to solve. That, however, is not
of much significance: what matters is that we believe we have
the equation which, in principle, is the correct equation for tin
description of Nature. (Actually that is not really true, for the
Schrodinger equation ignores relativity. Therefore it is only an
approximation, bearing a similar relation to a correct des
t 'While visiting Paris he (Victor Henri) received from Langevin a copv
of "the very remarkable thesis of de Broglie"; back in Zurich and having
not very well understood what it was all about, he gave it to Schrddinger,
who after two weeks returned it to him with the words: "That's
rubbish". When visiting Langevin again, Henri reported what
Schrodinger had said. Whereupon Langevin replied: "I think
Schrodinger is wrong; he must look at it again". Henri, having returned
to Zurich, told Schrodinger: "You ought to read de Broglie 's thesis
again, Langevin thinks this is a very good work"; Schrodinger did so and
"began his work".' Max Jammer (1966, p. 258).
cription as Newtonian mechanics bears to Einsteinian. This is
a difficulty which has been partly removed— see the Dirac
equation— but minor fundamental difficulties have in the past
bred cataclysm.)
The Schrodinger equation is a secondorder linear differ
ential equation in space coordinates (it contains terms such as
d 2 /dx 2 ] and a firstorder differential equation with respect to
time. Various forms of it are illustrated in Box 22. Written in
its full form it should be clear that It is not a wave equation,
for such an equation has secondorder derivatives with respect
to time. It may be regarded instead as a type of diffusion
equation; it is not unreasonable that the evolution of the
'wavefunction' in time should be akin to a diffusional process.
This point is of considerable significance, for the diffusional
form of the Schrddinger equation means that it is possible to
to interpret the wavefunction in terms of a probability of
discovering a particle in various regions of space (see Born
interpretation in the section on "wavefunction). Had the
equation been a true wave equation this interpretation would
have been untenable. The timedependence can often be
shaved off by the method of separation of variables (see
Question 2 and Box 22), and then we are left with the time
independent Schrodinger equation (see Box 22), which is of
the same form as an equation for a standing wave. It is from
this form of the equation that the name 'wave mechanics'
derives.
The timeindependent Schrodinger equation may be inter
preted as an equation for the curvature of the wavefunction,
and bearing this in mind enables one to anticipate some of the
features of its solution. The second derivatives 9 2 0/3x 2 , etc.
207
208
Schrodingcr equation
BOX 22: The Schrodinger equation
Timedependent form: Hty= ih0 */9 1 )
Wis the hamiltonian.
Timeindependent form: if Wis independent of time, ^ may
be written
#= if/ exp(i£ tfh),
where \j/ is independent of f and satisfies
Typical form of equation:
Onedimensional system; massm in a potential V{x\:
or
(f ) < f
+ l/(x}^U)=F0{x)
dx 2
V(jf)l^W = 0.
Threedimensional system; massm in a potential U(r):
V 2 #) + lW<r) = £^(r).
where V 2 is the "laplacian.
Separation of variables. Write ^ = >jjd, where i^ is a
function of position and a function of time. Let H be
independent of time. Then H^  ih\If becomes
0W0 = ih\#, or (1/^tf if» = ith/6)&.
The lefthand side is a function of position, not time, and if
x is varied the righthand side is invariant. Therefore l.h.s. =
constant = r.h.s.
Hence 6 = exp{i£f/h) and H\jj = E\j/.
are what we normally interpret as the curvature of the
function in elementary calculus, and we shall employ this
interpretation here. Note that when ip" fas we sna H denote the
second derivative) is positive the curvature of \p is like \. S ,
and when it is negative the curvature is like /"~N . The
essential feature to note is that the magnitude of the curvature
(the sharpness with which the curve bends) increases as the
total energy £ exceeds the potential energy V[x). This differ
E<V
E>V
y>>o
w
r\
1p<Q
r\
w
FIG. S1. Curvature of the wavefunclion at a point for different signs
of t}> and B— V at thai point.
ence £ — VM is just the classical kinetic energy at the
point x, and so we see that the "kinetic energy and the
curvature are proportional. Note too that the sign of the
curvature depends on the sign of the function \p itself: if E is
everywhere larger than V, the curvature has the sign of i> at
each point (Fig. S1). It is amusing to follow through the
implication of this for a free particle where V is constant and
E> V. Suppose we consider a point where \p> 0, then the
curvature is negative and so the function droops down towards
zero like /* — \ (Fig, S2). Sooner or later this droop causes it
to fall through zero and become negative. The hitherto droop
ing function acquires a positive curvature (because still E> V
but i/»< 0) and so begins to curl up towards the value zero.
This value it crosses and then again begins to droop back down.
The dependence of the curvature on the function therefore
forces the function to swing backwards and forwards across
the axis, and so to describe a harmonic wave. It is also amusing
to note that the rapidity with which it swings from positive to
negative values increases as E exceeds V: therefore the wave
length of the motion decreases as the kinetic energy increases
(see 'kinetic energy and the °de Broglie relation). If the
potential depends on position the wavelength is not a constant
(and not really defined}, but these arguments may be extended
to account qualitatively for the form of wavef unctions for
electrons in atoms and molecules.
Schrodinger equation
209
f
>0
r
FIG. S2. Propagation of a wave when E > V.
The only adequate way of accounting for the quantitative
form of the wavefunction in atoms and molecules is to solve
the equation mathematically. The immediate problem that
one encounters is that there exists an infinity of solutions for
a secondorder differential equation. The essential point at
this stage is the recognition that only some of the solutions
satisfy the stringent requirements of the Born interpretation:
when boundary conditions are imposed (that is, when one
states what conditions the function must satisfy at some
point of space) only a few solutions are acceptable. The
immediate consequence of this is that bounded systems are
quantized.
Quite often it is impossible to find analytical solutions to
the Schrodinger equation, or at least to find analytical
solutions that are not too complex to use. Under these
circumstances (which include the immensely important cases
of the structure of atoms and molecules) it is necessary to
resort to approximate methods: these include "perturbation
theory, "variation theory, and the method of "self consistent
fields. Timedependent perturbation theory is one way of
dealing with the Schrodinger equation. Some of the standard
solutions of the Schrodinger equation are discussed under the
appropriate headings (see particle in a square well, "harmonic
oscillator, "angular momentum, and hydrogen atom).
The fact that the Schrodinger equation is a linear differ
ential equation implies the validity of the "superposition
principle and all that flows from it.
Questions. T. What is the importance of the Schrodinger
equation? Why is the timedependent form not a true wave
equation, and how may it be manipulated into a form that
looks like a wave equation? What basic local property of the
wavefunction does the equation determine? Demonstrate
qualitatively and then quantitatively that the wavefunction
for a free particle is a wave of constant length. What is the
connexion between the length of this wave and the linear
"momentum and "kinetic energy of a particle described by
such a wave? Sketch the form of the wavefunction for a
particle of energy E in a potential field that decays linearly
with distance. Can a particle be described by a wave in which
at some point the energy E is less than the potential energy V
at that point? Sketch the form of a wavefunction for a particle
in which the potential rises linearly with distance (and crosses
the point V{x) = E). Sketch a number of possible wave
functions for a 'particle in a squarewell potential, and in a
"harmonic potential {try to do this without looking at the
answers, which will be found in the appropriate sections; if
you cannot do it, attempt to interpret the answers there in
terms of the discussion in this section),
2, Show, by the method of separation of variables, that the
timedependent equation can be separated into an equation
for the timedependence and an equation for the spatial
dependence, and solve the former. Proceed by attempting to
express the function Mx, f) as the product 0{t) \p{x), and
inserting this in the timedependent equation. Divide through
by &4j, and realize that one block of terms depends only on x
and the other depends only on t. Deduce that these two terms
must each be equal to some constant, which write E (on the
grounds that the constant has the dimensions of energy). Then
solve the equation for the time, and compare your answer with
that in Box 22.
210
second quantization
Further information. See MQM Chapter 2 for a way of setting
up the Schrodinger equation, and Chapter 3 for the method of
obtaining some of its solutions in a number of important cases.
A good account is also given in Pauling and Wilson (1935) and
in Kauzmann (1957). All books dealing with quantum theory
must, except for the most abstruse, mention the Schrodinger
equation. For Schrddinger's original papers see Schrodinger
(1926). See Chapter 5 of Jammer (1966) for a fascinating
commentary, and in an English translation in Shearer and
Deans (1928). The Schrodinger equation is only one of a
number of possible equivalent formulations of quantum
mechanics, and is not always the simplest to use, especially in
formal manipulations. Therefore see also matrix mechanics.
second quantization. Those who find quantization a
sufficiently difficult topic will be distressed to encounter
second quantization, and may have visions of a continuation
yet more subtle than Chinese boxes. But second quantization is
as far as things have gone, and is a device that enables one to
conjure elegantly with problems involving many particles,
including the problem of the electromagnetic field. It is a
mathematical artifice; but is that not possibly true of all
mathematical descriptions of Nature?
The mode of thought that leads to the introduction of the
idea is as follows. First quantization, which in our naivety we
have referred to elsewhere simply as quantization, replaces
observables by "operators, and the behaviour of a system, and
the results of experiments, are calculated by allowing these
operators to operate on the wavefunction obtained as a
solution of the "Schrodinger equation for the system. Thus
the dynamical functions have been replaced by operators
operating on a function \}/(x). But suppose by analogy with
this development we interpret the function \J/{x) as an
operator on something; then we have gone beyond our formal
procedure for quantization and are in the artificial realm of
second quantization. In this realm it is discovered that the
operator ^/*M operates (on something) to create a particle
atx, and that the operator \p{x') operates to annihilate a
particle at x'. It should be possible to appreciate that this
power of summoning and dismissing particles provides a means
of setting up equations that enable the quantummechanical
properties of manybody problems to be calculated. Thus
second quantization does not introduce a revolution into
physics (as did first quantization), but it does introduce a new
technique of calculation, and a new language— a language that
includes words such as photons, "phonons, "polarons,
"excitons, magnons, and rotons.
Further information. An introduction to the ideas of second
quantization will be found in a set of lecture notes by Atkins
(1973), which are based on a book by Mattuck (1967). Other
introductions of increasing sophistication will be found in
Ziman (1969), Roman (1965), Davydov (1965), Bogoliubov
and Shirkov (1959), and Schweber (1961). See also Kittel
(1963).
secular determinant. In the construction of a "molecular
orbital by the method of "linear combination of atomic
orbltals one attempts to express the orbital as a sum
c ty + c \p + . . ., where the i^s are the atomic orbitals on
atoms a, b, etc. and the cs are the coefficients to be modified
until the best set of values is found: this is done by seeking the
set that gives the lowest energy. Instead of idly toying with
random different values of the cs until mental decay super
venes, it is desirable to have a short, sharp method for finding
the best set, and this is the role played by the secular
determinant.
In the Questions you are asked to show that the best com
bination of orbitals is given by one of the solutions of a set
of simultaneous equations in which the coefficients c are the
unknowns (see Box 23). These are the secular equations. In
common with other sets of simultaneous equations they
have a nontrivial solution when the determinant of the
factors of thecs disappears (the trivial solution corresponds
to all the cs themselves vanishing). This determinant is the
secular determinant. See the Box. The secular determinant
vanishes for N values of the energy of the system, if there are
N atomic orbitals contributing, and the lowest roof may be
identified with the lowest energy of the system, that can be
attained on this model. The values of the cs corresponding to
this root may be found by the normal methods (brute force,
or intelligence via the method of cof actors, see Box 23) and
give the best wavef unction for the system. The other (A/— 1)
roots may be identified with higherenergy orbitals. for the
secular determinant
211
BOX 23: The secular determinant
Direct approach. Write the Schrodinger equation H\p=&ip
in terms of the expansion
fit
y= 1
and obtain N N
£y*/£ «*.*. ft
y  i /  1
Multiply from the left by \p. and integrate, to obtain
N
£<y[H&S]= (/=1,2,.../V) (2)
/i
with H j}  fdTfyhtty and S /y = fdT\j/.\L. This is a set of /V
simultaneous equations for the coefficients c, and possesses
nontrivial solutions when
A= det l/A.
8
es y i = o.
(3)
Expansion of this fl/X/V determinant, and solution for £
leads to the N eigenvalues of H, In general, if N is finite the
eigenvalues are only approximate.
To each root & of {3} there corresponds a wavef unction ^
expressed as in (1 ). To find the coefficients c. use Cramer's
rule. (See Margenau and Murphy {1956, p. 313) or Irving
andMulfineux(1959,p. 269).) Let A, = c./c,; then (2)
becomes
iLkfVHfi ~ &S v l = m ft  H n ) (/ = 2, 3, . , . N\.
Let D be the (Af— 1) X (A/1 ) determinant formed from
this H l}  &S.j, and D w {h) be the determinant formed by
replacing the nth column in D by (£S 2I  tf 2l , &S 3 ,  W 31
. , , &S
— MffJ Then according to Cramer's rule:
., c N /ci. To complete the
This gives the ratios c 2 /ci ,
determination use
c i + c\ +
..+41.
(4)
This procedure gives the coefficients e. corresponding to
the root &, and should be repeated for each root of (3).
Variational approach. Minimize S with respect to variations
in the coefficients c,
The condition (3S/c)c.) = is satisfied when
N
sty
(5)
A =
(Confirm this by substituting (1), taking the derivative of £,
and using simple algebra.) This is identical to (2) and the
procedure is as before. The lowest of the N roots of (2) is
the minimum value of £.
Example. Suppose N — 2, then the secular equations are
Ci Wn  &S n ) + c 2 {H n  &S n ) = )
ci (W 2I  &S 2 i) + c 2 (W 22  £$ 22 » = \
and the secular determinant is
"31 ~~ &S 2 1 H22 — &S22
The roots of A are the roots of
<S U S 22 S 12 S 21 )g 2 + tH,A, + H 2l S n  H 22 S n  «„S 22 )E +
and the lower is the minimum of S. The coefficients are
determined from D = W 22 — &S 22 and eqn (4):
«2 W 22 — &S22) — ~ {H2I ~ SSji)
and c ^+c = 1.
In the special case when S n => S 2 2 = 1; Si 2 = S 2i = the
solution may be put into the form
« _ j Wn — H n cot0 ip = i^,sin^  \p 2 cos8
\ rin + H l2 coxd \p = 1^ cos0 + ^ 2 sin '
where = ^arctan[2W, 2 /(W 11 W M )].
molecule, and the corresponding coefficients give the other
(/V — 1 ) molecular orbitals.
The secular determinant crops up wherever one has several
orbitals which one anticipates may interact. If there is a
'perturbation that can mix one state with another the true
212
selection rule
ground state of the system is best found by a linear sum of
the two unperturbed states. The energy of the new system is
given by the lowest root of the secular determinant, and the
set of coefficients corresponding to that root gives the best
modification of the wavefunction. One example of this type
of situation is provided by •configuration interaction, and
another by the Huekel method of conjugated molecules.
The name secular originates in the appearance of the same
kind of determinant in classical mechanics, and especially in
celestial mechanics. A secular variation in the motion of a
body, and in particular the orbit of a planet, is one that
gradually develops over a long period of time {saeculum:
Latin for age or generation) as opposed to one that varies
rapidly or periodically and is not cumulative. Why on earth,
if you will forgive the allusion, does this have anything to do
with molecular structure? The answer lies in the fact that
variation theory may be related to perturbation theory, and
that the perturbations of interest to variation theory are
those that accumulate to give rise to a set of orbitals with
welldefined and constant separation. This shows up
especially clearly in degeneratestate perturbation theory, but
that is a subject that deters unless it is called by some other
name. One name that ought not to deter is "molecularorbital
theory, for when homonuclear systems are considered the
molecularorbital method is equivalent to degeneratestate
perturbation theory. This can be appreciated by realizing
that in the absence of interaction between the atoms all the
atomic orbitals {of the same quantum numbers) which later
are to be combined have the same energy (are degenerate).
The effect of the interatomic interaction is like a perturbation
on the degenerate systems. If we imagine putting the atoms in
a molecular conformation, forbidding interaction, and then
gradually turning on the interaction, we can appreciate that
the perturbation gradually accumuiates and the different
linear combinations of atomic orbitals diverge until they
attain the separations characteristic of the molecule. Thus we
are really considering a strong secular perturbation on the
atomic orbitals, and the separation of the levels can be found
by the application of the secular determinant.
The language of ordinary "perturbation theory also draws
on the word secular. The expression for the energy to second
order in some perturbation consists of a part involving only
the original state of the system (see Box 16 on p. 172) and a
part involving "virtual transitions to excited states. The former
terms represent the effect of a secular term, or secular
perturbation, whereas the latter represent the effect of the
nonsecular terms.
Questions. The variation method considers a linear com
bination of orbitals of the form i/i = c a ^ a + c b ^ b for a two
orbital system and then calculates the minimum value of
& = JdT^H\p/fdT\}j\p. Show that the extremal values of this
expression correspond to the solutions of tfie two simultaneous
equations W 3o  &SJc a + («„„  S>SJc b  and
are various integrals. Set up and solve the secular determinant
for this problem, and find expressions for the two sets of cs
corresponding to the lower and higher energies. Apply this
calculation to H 2 by identifying ^ a and ip b with Isorbitals
on nuclei a and b respectively, taking S aa = S bb = 1 ,
H = W hh = a, and H . = tf h = j3. Suppose that S = 0.
aa bb lib ba . lM
Now generalize the calculation to an orbital of the form
c \1/ +c \b +... c„ 0„ and show that the solution of the
a T ab T b IV Iv
variational problem leads to an N X N determinant. Approxi
mations are usually made as to the values of the integrals
involved in these secular determinants: this is the realm of the
Huckel method and its analogues, and you will find more
problems set there.
Further information. See MQM Chapters 7 and 10; the former
deals with degeneratestate perturbation theory and the latter
with the application of the secular determinant to various
aspects of molecular structure. Further details will be found in
§9.6 of Coulson (1961), §2.1 of Streitweiser (1961), §Vl.24
of Pauling and Wilson (1935), §7band§11bof Eyring,
Walter, and Kimball (1944), Pilar (1968), and McGlynn,
Vanquickenborne, Kinoshita, and Carroll (1972). See Huckel
method, 'configuration interaction, and "perturbation theory.
selection rule, Spectral lines result when a system makes a
transition from one state to another of different energy. All
lines in a spectrum can be related to the difference between
the energies of states of the system (each line can be expressed
as a combination of "terms), but not all possible pairs of states
give rise to spectral transitions: some transitions are allowed
and some are forbidden. Selection rules tell us which are
allowed and which are forbidden. They are generally quoted
in terms of the changes that may occur in a "quantum number,
but sometimes they are rules about the way that the symmetry
of the state may change. Occasionally one encounters the term
gross selection rule: this refers to a property that a molecule
must possess in order for the remaining selection rules to come
into operation. Selection rules for a variety of transitions are
shown in Box 24.
There are various ways of understanding why the selection
rules govern as they do. The gross selection rules refer to the
fact that the molecule must have some way of interacting with
the electromagnetic field: the presence of a permanent dipole,
for instance, means that the electric field of a passing light
beam may accelerate the molecule by exerting a torque and so
induce a rotational transition. The other selection rules can
generally be understood in terms of the possession by the
photon of a definite intrinsic angular momentum (a photon
has spin,. On absorption the photon is annihilated but
angular momentum must be conserved, and this momentum
appears in the electrons or in the nuclear framework. Rules
selection rule
213
BOX 24: Selection rules
Atoms
Electric dipole transitions:
A/ = 0,±T;butJ = 04*J =
9— >u;g4*g, u4+u (Laporte rule)
AS=0
4t = 0,±1;buti = 0ki = o
Magnetic dipole transitions:
A/ = 0, +1; but J = 04*^ =
g — *g, u — >u;g+>g
AL = 0, ±2.
Electric quadrupole transitions:
A/=0,+1,±2;butJ = 04*./=0
g— *g, u— >u;g4*u
AL = 0,+V . ±2; but L = 4* L = 0.
n /.^coupling.
>A =0:
Molecules
Electric dipole transitions:
electronic transitions
AJ=0, ±V,butJ = Q+>J =
+ *; + 4*+,4*
g— *u;g4*g, u4*u
s — * s, a — * a; s 4+ a
case (a): AA = 0, ±1; but for A=
T— *!?, 2T
AS=0
A2=0
cases {a} & (c): AJ2 = 0, ±1; but 12= 04* £2 = if AJ =
cases (b) & (d): AN = 0, ±1; but in (b) A= 04* A=
for AN = 0.
vibrational transitions:
absorption (i.r.): dipole moment must change along
"normal coordinate
Af=±1.
Raman: polarizability must change with vibration
Ap=±\.
vibrationrotation transitions:
+2 Sbranch (Raman, Stokes)
+ 1 Rbranch (i.r.)
Qbranch t (i.r.)
'—1 Pbranch (i.r.)
[ ~2 Obranch (Raman, antiStokes).
t Only if molecule has component of
a.m. about axis.
rotational transitions:
absorption (microwave): molecule must possess
permanent dipole moment
Aj=+1, AK =0.
Raman: anisotropic polarizability of molecule
Aj = ±2 (Stokes: +2; antiStokes: —2).
Ac = ±1; AJ =
214
self consistent field
such as M = ±1 or AJ = ±1 reflect this conservation of
angular momentum, and are discussed further under 'electric
dipole transitions and in the sections on the different types
of spectra (see also "magnetic dipole and electric quadrupole
transition for a variant of this rule).
AH the rules may be deduced from an examination of the
form of the transition dipole moment (see electric dipole
transition). One very important way of examining this
moment is by "group theory: those who know about group
theory and irreducible representations should remember that
the product of the irreducible representations of the initial
state, the final state, and the transition moment operator
must contain the totally symmetric irreducible representation
for the transition not to be forbidden (Box 4 on p. 33). Those
who do not know enough group theory to understand this
important rule should learn some as soon as possible.
Further information. See MQM Chapter S for a discussion of
the grouptheoretical basis of selection rules, and Chapters
8 and 1 for their application to all the types of atomic and
molecular transitions. Other group theoretical discussions
will be found in the references in the entry on group theory.
The applications to spectra are also described under
"electronic, vibrational, rotational, and "Raman spectra, and
detailed accounts will be found in Barrow (1962), Whiff en
(1972), King (1964), and Herzberg (1944, 1945, 1950, 1966).
self consistent field. The selfconsistent field (SCf ) method
of calculating atomic and molecular structures was originated
by Hartree, improved by Fock, and used by almost everyone.
The basis of the method is to guess the °wavef unctions for all
the electrons in the system. Then one electron is selected and
the potential in which it moves is calculated by freezing the
distribution of all the other electrons and treating them as the
source of the potential. The Schrodinger equation for the
electron is solved for this potential, and so a new wavefunction
for the electron is obtained. This procedure is repeated for all
the other electrons in the system, using the electrons in the
frozen "orbitals as the source of the potential. When the cycle
is completed (Fig. S3) one possesses a set of orbitals for all
the electrons of the system, and in general these will differ
from the original, guessed set. Now the cycle is repeated, but
improved wavefunctions generated by the first cycle are used
as the initial guess: a complete cycle generates a new set of
improved functions. This sequence is continued until passing
a set of orbitals through the cycle leads to no change: the
orbitals are then self consistent.
Storf here
FIG. S3. The self consistency
cycle.
Bid of game
self consistent field
215
1. The Hartree method takes the view that the atom or
molecule can be represented as a product of orbitals, one
for each electron. Therefore it guesses a set of functions, and
sets the selfconsistent machinery in operation; after a number
of cycles the solution has become stable, and so one has a
function (still a simple product of oneelectron orbitals) that
describes the structure of the system, and a set of orbital
energies from the solution of the Schrodinger equation. The 1
calculation is actually done with the Hartree equations (see
Box 25) in which the potential experienced by one electron is
BOX 25: Selfconsistent fields
Hartree equations
where the Coulomb operator is
/ i
1
J f m ~ /dr 2 4** (2) $. (2) (eV47reor 12 ).
Sum/ runs over occupied orbitals.
HartreeFock equations (closed shell)
[W corB (1) + 2j>.(1) £ #f (1)] tyi, = e^.m,
where the exchange operator is defined through
*,■ ( 1 ) lf», < t ) = Jdr 2 tyf (2) }jj. m tftAiKtfn) tym
With J,y = /dT, ^'(1)^(1 ) 1^.(1}
as the Coulomb and exchange integrals.
the sum of its interactions with all the other electrons. The
presence of the electron is not allowed to distort the electron
clouds locally: its effect is to distort the orbital as a whole. No
possibility of the other electrons tending to keep away from
the position of the electron of interest is admitted: electron
correlation effects are ignored; see Fig. C3 on p. 43.
2. I n the HartreeFock method the selfconsistent field
procedure takes into account the indistinguishability of the
electrons as required by the Pauli principle. That is, it allows
for the possibility of "exchange. To do so it takes as the
wavefunction a 'Slater determinant, and then enters the
selfconsistency cycle of Fig. S3. The potential experienced by
each electron is the Coulombic potential modified by the
"exchange energy {which is the correction of the Coulombic
repulsion energy required in order to take into account the
tendency of electrons with like spins to avoid each other). The
Schrodinger equation is cast into the form of the HartreeFock
equations (Box 25), where the first part of the potential is the
uncorrected Coulombic interaction of the electron with all the
other electrons in their frozen orbitals, and the second part is
the exchangeenergy correction. The HartreeFock (HF)
method also neglects the correlation energy. The unrestricted
HartreeFock (UHF) method allows more freedom to the form
of the orbitals by permitting the spatial form of the orbital to
depend on whether the electron has an a or a (3 "spin.
It is normal for the atomic and molecular orbitals used as
a starting point in SCF calculations to be linear combinations
of atomic orbitals, and the accuracy of the calculation is
severely curtailed if the functions chosen are too inflexible;
this might occur if too few (too small a basis set] have been
chosen. A convenient set of orbitals with which to commence
a calculation are the "Slater atomic orbitals. The evaluation of
molecular integrals is considerably simplified if gaussian
orbitals are used, but more of these must be used if the atomic
wavef unctions are to be at all reasonably represented. When
the labour of an SCF calculation appears to be too great, or is
actually found to be too great, approximations are introduced
in a more or less rational fashion: such methods constitute the
semiempirical SCF calculations (see Huckel method) as
opposed to the a priori or 'ab initio SCF calculations which
begin from scratch and proceed without approximation (apart
from the approximations inherent to the HF scheme). In all
cases the calculations can be improved by permitting con
figuration interaction.
Questions. Outline the sequence of calculations involved in a
selfconsistent field calculation. When does the cycling pro
cedure cease? What is the basis of the Hartree method? What
does it neglect? What is the basis of the HartreeFock method,
and why is it an improvement on the Hartree method? Why is
216
sequence
the HFSCF method unable to provide exact atomic wave
functions and energies? What is the UHFSCF method? What
interpretation could be put on the letters HF— SCF— LCAO—
MO, and what process of computation would you understand
by it? What is the difference between an ab initio SCF cal
culation and a semiempirical calculation? What errors are
introduced by using too small a basis set, and what is meant
by a 'basis set' in this context? Show that it is necessary to
deal with electron repulsion and exchange integrals involving
orbitals located on up to only four atomic centres. Why is it
unnecessary to invent methods to deal with 5centre integrals
in HF calculations?
Further information. See MQM Chapter 8. A simple intro
duction to the methods of atomic and molecular energy
calculations is given by Richards and Horsley (1970) and by
McGlynn, Vanquickenborne, Kinoshita, and Carroll {1972}. A
classic account of atomicstructure calculations, which
illustrates the headaches which Hartree must have suffered
before electronic computers were available, is described in
Hartree (1957), Both Richards and Horsley and McGlynn
etal. work through sample calculations. The semiempirical
methods are described under "Huckel method.
sequence. First review and be quite clear about the meaning
of a "progression in the "vibrational structure of the "electronic
spectra of molecules. A sequence is a series of lines that have
in common the same value for the difference of the vibrational
quantum numbers for the upper and lower electronic states. If
the upper vibrational quantum number is V and the lower V ,
then the lines that have v' — v" = form one sequence, those
with V — y" = —1 form another, those with j/ — 1>" = + 1 a
third, and so on (see Fig. S4).
All the lines of one sequence would lie at the same
frequency if the vibrational energy levels in both the
electronic states were evenly spaced, but the anharmonicity
of the vibrations destroys the even separation of a truly
"harmonic oscillator, and so the separation of the lines of a
sequence yields information about the deviation of the
molecular potential energy curve from an ideal parabolic
form.
5 */
6u=\
4
3
2
I
6v0
V
FIG. S4. Two sequences.
Questions, What is a sequence, and how does it differ from a
progression? What information is it possible to extract from
the positions of the lines in a sequence? When may all the lines
of a given sequence be coincident? Would you expect the
intensity of the lines in a sequence to be the same; if not, what
is a major influence on the intensity? Given that the vibrational
energy of the ground state depends on v" through the
expression gj'V + fl + x"jJ^ + \) % . with a similar
expression for the excited electronic state vibrational energies,
calculate the frequencies of the 0, — 1, and +1 sequences.
Furtfier information. See MQM Chapter 10. Detailed infor
mation about the appearance, analysis, and formation of
sequences will be found in Barrow (1962), Whiff en (1972),
Gaydon [1968), and Herzberg (1950, 1966).
series. The frequency of any spectral transition may be
expressed as the difference of two "terms, each term
representing the energy of a state of the atom or molecule.
Let us denote the terms T , where n is some index which
is generally identified as a "quantum number or a collection
of quantum numbers for the system. The frequency of each
transition from a state n to a series of other states n' is simply
~^n> ~ T n' and tne series of " nes ' m tne spectrum for a fixed n
and changing n is called a spectral series.
Some of the most famous spectral series occur in the alkali
metals and involve transitions of the single valence electron. The
transitions in which an electron in a porbital drops down into
the groundstate sorbital (see Fig, S5) gives a series of lines
known as the principal series (and hence the notation 'p' for
the orbitals involved}; the series formed by the light emitted
as the electron falls from some upper sorbital into the lowest
porbital constitutes the sharp series {and hence V); the
decay of electrons from the upper excited dorbitals falling
into the lowest porbital gives a diffuse series {thus it looks,
u
S
I
o
5000
15000 t
25000.9
s
2
iSOOO
FIG. S5. S, P, D, F series in sodium.
singlet and triplet states 217
and hence 'd'); and as electrons in forbitals drop to the
lowest dorbital so is generated the fundamental series (and
hence T). The transitions are illustrated in Fig. S5.
Further information. See MQM Chapter 8 for more infor
mation about series. The structure of atomic spectral
series is described by White (1934), King (1964), Herzberg
(1944), Whiffen (1972), Kuhn (1964), and Condon and
Shortley (1963). The "selection rules that led to the con
struction of Fig. S5are described in that section. Molecular
series are discussed in King (1964), Barrow (1962), Whiffen
(1972), and Herzberg (1950, 1966).
singlet and triplet states. In a singlet state the net 'spin
of a manyelectron system is zero {S= 0). In a triplet state
the net spin is unity {S = 1). The spin "angular momentum
•vector may have a series of projections on a selected axis.
These projections are distinguished by the quantum number
M s which can range in unit steps from S down to S. It
follows that /W„ may take on three values (M„ = 1, 0,1)
o o
when S = 1, but only one value {M s = 0) when S = 0: hence
the names triplet and singlet.
The distinction is easiest to see in the case of a system
composed of two electrons. As each electron can have a pro
jection m s = +j or — j (which we denote a or 0) the com
bined system can be in any of the four states a(1 )u{2),
<*{1 )0{2), £(1 )a(2), and j3( 1 )j6(2). The middle two choices do
not correspond to a resultant spin vector of fixed length
because the a and vectors can make any azimuthat angle to
each other. If we specify the azimuth of one with respect to
the other we shall get a definite resultant: if a{1) and (3(2)
are in phase they give a resultant corresponding to S = 1; if
they are 180° out of phase their resultant is zero and
corresponds to a state with S = (see Fig. S6). It is worth
emphasizing that when we say that spins are 'paired' in a
singlet state we mean not only that one has a spin and the
other but also that they are relatively oriented so that they
point in opposite directions. (In the M s = state of the triplet
one electron has a spin and the other )3 but their resultant is
not zero.) One can show from quantum mechanics that the
appropriate form of the spin function for the triplet (inphase)
state with M s = is a{1 )(3(2) + 0( 1 )a(2), and for the singlet
218 Slater atomic orbitats
FIG. S6. Triplets and singlets:
note the different phase of a and
in the M =0 state of the singlet
and the triplet.
(outof phase) state it is <*(1 )0(2) — 0(1)0(2), The other two
states of the triplet (M s = ±1) area(1)a(2) and j3< 1)j3(2).
It is possible to convert a singlet term into a triplet (and
vice versa} by making one electron spin precess faster than the
other. This may be brought about by applying different mag
netic fields to the two spins so that their Larmor frequencies
(see "precession) differ: an inphase (triplet) orientation
is thereby gradually turned into an outofphase orientation
(Fig. S7). A field from a laboratory magnet cannot effect the
interconversion (which is known as intersystem crossing, ISC)
because it affects both spins equally {the magnet's field is
homogeneous on a molecular scale). A magnetic field arising
from within the molecule may be able to rephase the spins.
For example, the "spinorbit coupling to one electron might
FIG. S7, Relative rephasing of the spins lead to tripletsinglet inter
conversion. The hatching denotes a region of a different spinorbit
coupling.
differ from that to the other, and as this interaction is
magnetic, we have a situation in which the Larmor frequencies
differ. This is the reason why molecules containing heavy
atoms (with large spinorbit couplings) are efficient at inter
system crossing (see "phosphorescence).
The interconversion of ortho and parahydrogen (see
°nuclear statistics) can be brought about by paramagnetic ions
because the singlet and triplet phasing of the nuclear spins can
be interconverted by the inhomogeneous field which such an
ion may generate if it is closer to one nucleus than the other.
Further information. See Chapter 6 and Chaper 10 of MQM
for a more detailed discussion of singlets and triplets, and their
interconversion. The role of the triplet state in chemistry, and
its detailed quantum mechanics, is discussed in McGlynn,
Azumi, and Kinoshita (1969), who give many references. The
photochemical consequences of the differences between singlet
and triplet states are discussed by Wayne (1970) and Calvert and
Pitts (1966). The difference is taken up in Hund rules and
"spin correlation, which you now should see. A manifestation
of singlettriplet ISC is "phosphorescence.
Slater atomic orbitals. Atomic orbitals in manyelectron
atoms have a complicated dependence on position which can
be represented accurately only be listing their amplitude
numerically. For many purposes it is desirable to have an
analytical function rather than a table of numbers, and the
Slater determinant
219
Slater atomic orbitals are analytical functions based on the
numerical results, but designed to reproduce them with
moderate accuracy. A set of simple rules has been devised
which enable the Slatertype orbital (STO) to be written for
any electron in any atom.
Each orbital has a radial dependence given by Mr"' " 1
exp(— £r) and an angular dependence given by the "spherical
harmonics corresponding to the appropriate values of the
quantum numbers E andm £ (see Table 23). The rules for
finding the effective principal quantum number n* and f are
as follows.
t. n* is related to the actual principal number n by the
correspondence n •— > n* using the rule 1 — » 1,2 — *■ 2,
3 * 3, 4 — > 37, 5 + 4, and 6 * 42.
2. {" is related to the effective atomic number Z _ by
i  % ff '' 1 > z eff is related to the true atomic number by
^gtf — Z — a; a is the screening constant.
3. The screening constant is calculated by classifying
atomic orbitals into the following groups: (Is); (2s, 2p);
(3s, 3p); (3d}; (4s, 4p); (4d); <4f); (5s, 5p); <5d); . .'..tristhe
sum of a number of contributions arising from each group, and
is calculated as follows. Let the atomic orbital of interest be in
a group X and let it have a principal quantum number n. The
contribution from the other electrons present is
(a) from electrons outside group X (that is, to the right of
X in the list): 0;
(b) from electrons in group X; 030 if the electron is Is but
035 from any other electron in X;
(c) if the electron of interest is/is or np
(i) for each electron with principal quantum number
number n— 1: 085;
(ii) for each electron with principal quantum number
n2, n3, . ..: 100;
(d) if the electron of interest is nd or of, for each electron
in a group preceding X in the list: 100.
Slatertype orbitals for the valence orbitals of the firstrow
atoms are given in Table 21.
Inspection of the form of the Slater orbitals reveals a
serious defect: they possess no radial nodes. One consequence
of this is that they are not "orthogonal. They may be made
mutually orthogonal by the Schmidt orthogonal ization pro
cedure, and so this defect can be overcome (see "orthogonal
functions for the procedure).
Questions. Deduce the form of the Slatertype atomic
orbitals for the Isorbital in H, 2s in Li, 2s and 2p in C, N, O,
and F, and the 3d in Fe 2 * and Fe. Find the normalization
constant for the general STO. Find the mean radius of the
electron distribution in each of the orbitals just set up. Show
that it is possible to choose a sum of the 1s and 2s STOs that
is orthogonal to the Isorbital: this is the Schmidt procedure
for orthogonal izing 2s to 1s, and it may be extended to
orthogonal ize 3s to both Is and 2s: do so (see "orthogonal
functions). What effect does orthogonal ization have (in this
case) on the number of radial nodes, and for the C atom, the
mean radius of the 2sorbital?
Further information. See MQM Chapter 8 for a brief dis
cussion. A useful discussion of Slatertype orbitals is given
in §11.8 of Coulson (1961), Murrell, Kettle, and Tedder (1965),
and McGlynn, Vanquickenborne, Kinoshita, and Carroll
(1972). The last, in Appendix B, give many references to the
expression of "selfconsistent field orbitals in terms of sums of
STO's and a table of orbitals. For the Schmidt orthogonal
ization see "orthogonal functions. "Overlap integrals involving
Slater atomic orbitals are referred to under that heading.
Slater determinant. According to the "Pauli principle the
wavef unction for a system of electrons must change sign
whenever the coordinates of any two electrons are inter
changed. It follows that a simple product of the form
0f <1}y^ (2) i^(3) ... i^(W), where electron 1 occupies
orbital \j/ g with spin a, and so on, is inadequate. It is possible
to ensure that a product of this form does satisfy the Pauli
principle by writing it as a determinant:
(1W«) V4
<<D ^ a (2) ^(3)
Km *J<2) tffoi
^(1) ^(2) ^ a (3)
1#U) ^(2) ^(3)
i/£(/V)
$*m
220
spectroscopic perturbations
Expansion according to the rules of manipulating determinants
leads to N\ terms, half occuring with a +ve sign and half with
ve. The factor {1//V!} M ensures that the determinantal wave
function remains "normalized. That this Slater determinant
satisfies the Pauli principle follows automatically from the
property of determinants that interchange of any pair of rows
or columns reverses its sign. Suppose that we interchange
electrons 1 and 2, so that electron 1 is put into the orbital
hitherto occupied by electron 2, and vice versa. The effect on
the determinant is to interchange the first and second columns,
and so the sign changes. The same happens when any pair of
electrons are interchanged, and so the determinant is the
appropriate combination of the oneelectron orbitals.
It should be observed that the "Pauli exclusion principle
follows from the disappearance of a determinant when any
two rows or columns are identical. Suppose that electron 1
entered orbital \p with spin a and that electron 2 joined it
with the same spin. Then the first two rows of the determinant
would be the same, and so it would vanish; therefore it is not
possible to form a state in which more than one electron
occupies the same orbital with the same spin.
A word on notation: the orbitals with their accompanying
spin are known asspinorbitafs. A spinorbital corresponding
to spin « instead of being written i^* is sometimes written
merely t/> with the a spin understood. In this notation the
p 1 spinorbital is denoted $ . Much paper would be employed
if a determinantal wavef unction were always written in full;
therefore it is normally denoted by listing only the terms on
the diagonal and ignoring (but remembering) ^normalization
constant. The determinant above becomes l^ a V i^ b   ■ V z '
in this notation.
It should be noted that only for closedshell species can the
wavefunction be represented by a single Slater determinant;
when the shell is incomplete a linear combination of determin
ants must be used.
Questions. Why is a simple product of orbitals an inadequate
representation of the state of a manyelectron system? Why is
a Slater determinant a suitable representation? Write the Slater
determinant for the helium atom, expand it, and confirm by
inspection that it satisfies the Pauli principle. Do the same for
four electrons in the lowestenergy configuration of a one
dimensional square well. Confirm that the helium atom must
have paired spins in its ground state, but that in an excited
state they may be paired (a singlet) or unpaired (a triplet).
Repeat these considerations for the hydrogen molecule.
Continuing with a twoelectron system, write down the
"hamiltonian and show that a simple (nondeterminantal)
product function leads to an expression for the energy in
which the electronic interactions are represented solely by a
Coulombic repulsion term, but that when a Sister determin
ant is used an additional integral appears (the exchange
integral). What is the value of this exchange integral in the
ground state of helium?
Further information. See MQM Chapter 8. See Richards and
Horsley (1970) for a gentle introduction to the way of manipu
lating determinantal wavefunctions, and McGlynn,
Vanquickenborne, Kinoshita, and Carroll (1972) for a more
detailed version. The role of the Pauli principle in determining
the energies of atoms and molecules is described under
Pauli principle, and aspects of the consequences are the
■exchange energy, Fermi hole, and spin correlation. The
•localization of molecular orbitals into regions of a molecule
can be demonstrated in terms of properties of a Slater
determinant.
spectroscopic perturbations. The presence of a
perturbation generally muddles a system by shifting energy
levels and causing states to take on to some extent the
characteristics of other states. Spectra arise from transitions
between energy levels, and therefore perturbations appear as
shifts and changes in intensity of the spectral lines. For
example, one might be following a series of spectral lines
forming a regular array on a photographic plate, and then in
the region of a particular frequency the lines lose their
regularity and the spectrum seems distorted. This is a
spectroscopic perturbation and has arisen because several
states that can be mixed by a perturbation have come close
together in energy.
Two classes of perturbation are often distinguished: a
homogeneous perturbation is an interaction mixing vibrational
and electronic levels, and in a linear molecule it mixes states
with the same value of the quantum number A. fKhetero
spherical harmonics 221
Aeneous perturbation is a rotational electronic interaction
(as in A doubling) and mixes states differing in A by
±1.
Further information. See §6.19 of King (1964) and §V.4 of
Herzberg (1950). See also Herzberg (1966), Kronig (1930),
and Kovac's (1969). Revise perturbation theory, 'super
position principle, "resonance, and 'predissociation.
spherical harmonics. The spherical harmonics are a set of
functions of the angular coordinates Q and (as defined in
Fig. SS), that satisfy the differential equation A T Yn (0, 0) =
~~ W + ""^£^(0, 0), where A 1 is the legendrian operator
(see 'laplacian). These may be expressed as simple poly
nomials of sin 8, cos $, sin 0, and cos 0, or as polynomials in
x, y, and z (Tables 22 and 23), and visualized as the vibrational
modes of a spherical shell.
colofifude
FIG. S8. Spherical polar coordinates.
Each function is distinguished by the labels £ and m: K may
take any positive integral value, including zero; and m, for a
given value of 8, may vary in unit steps from — £ to B. These
properties of £ and m should be strongly reminiscent of the
properties of "orbital angular momentum, and it is in fact the
case that the spherical harmonics are the wavefunctions
corresponding to states of angular momentum H,m.
The function V o , o [0,0) is a constant (1/2/ 7r) and therefore
has the same value at all points on the surface of the sphere: it
can therefore be depicted by the function drawn in Fig. S9 a,
where V , is the surface at a constant height above the
spherical surface. A convenient way of denoting this isotropy
spherical
harmonic
auxiliary
[uiH_rlcn
nodes
FIG. S9. (a) /oo(0, 0), its auxiliary, and a crosssection. bj V ln W, <b\.
M Yx>iO. 0).
(or spherical symmetry) is to mark off the value of the
function at a point as a length along the radius of the sphere:
this gives rise to an auxiliary function, itself a sphere of radius
y<j.o. which contains all the information about the shape of
V ,o itself. The connexion between this auxiliary function and
the representation of a Is 'atomic orbital should be noted
{see p. 108). There are no angular 'nodes in Y 0l0 . and so it
corresponds to a wave with zero "orbital angular momentum.
222
spherical harmonics
The function Y ll0 [8, <j>) is proportional to cos and is
independent of <p. It has extrema at the poles and is zero
on the equator, and may be represented as in Fig, S9 b. The
auxiliary function may be constructed in the same way as
before, and it will be observed that in the northern hemisphere
the wave has a positive amplitude, and that this is denoted by
+ in the auxiliary function; and that in the southern hemi
sphere the amplitude is negative. The resemblance of the
auxiliary function to the boundary surface of a p^orbital
should be noted. Since there is one angular node the wave
represents a state of angular momentum with £ = 1. An
observer looking along the zaxis would not see the equatorial
node, and so the component of momentum about the zaxis
is zero (and that, it should be recalled, is the significance of
the quantum number m being equal to zero).
The other spherical harmonics corresponding to £ = 1 are
those with m = 1 and — 1 : both these functions are complex
(but not complicated) and correspond to waves running round
the zaxis with their amplitudes predominantly in the
equatorial zone. Combinations of the two functions may be
constructed that are real, and correspond to standing waves:
one combination is Y ul + Y lt _ u which is proportional to
cos $ (inspect Table 23). This has the same shape as / ,, but
is orientated along the xaxis. The combination Y il} — Vi,_i,
which is proportional to sin 0, is also of that shape and is
directed along the jeaxis. The connexion of these combinations
with the p  and the p orbitals should be appreciated.
The procedure may be continued to include the five functions
with £ = 2, and the functions generated have a close
resemblance to the dorbita!s of atomic theory (Fig. S9 c
and Fig. H12on p. 105). In each case the number of angular
nodes is equal to £ (we count each plane surface as a node) and
the connexion of the number of nodes and the angular
momentum arises as a consequence of the e de Broglie relation
and the connexion between the "kinetic energy of a state and
its wavefunction. The orbital shapes and the atomicorbital
boundary surfaces are not identical, because the latter are
boundaries enclosing chosen amounts of amplitude or density
and the dependence of these on the radius (as well as the
angles) distorts the spherical harmonic shapes.
If it is desired to find the distribution of a particle sliding
round the surfece of a sphere with a particular angular
momentum, then in accord with the Born interpretation (see
wavef unction) it may be calculated by taking the square of
the appropriate function. Thus a particle with zero angular
momentum [S ■ 0) is spread evenly over the surface; one with
£ = 1 is found either predominantly in the polar regions
[m = 0) or in the equatorial (m = ±1); one with £ = 2
is found strongly in the polar regions, but also with a significant
density spread smoothly around the equator (m = 0), or in
two bands concentrated at 45° N and 45° S (m = ±1), or
in a band highly concentrated on the equator [m — ±2).
Questions. 1. What is the significance of the spherical
harmonics in classical theory and in quantum theory? What
values may the indices £ and m take, and how many values of
m are permitted for a particular value of £? What is the shape
of V ,o, and why is it plausible that it corresponds to a state of
zero orbital angular momentum? To what vibration of the
surface of a sphere does it correspond? Construct the
auxiliary function for a p^, a d^.i, and a d orbital. Draw
the shape of the function that determines the probability
distribution of the particle in each of these orbitals. To what
vibrations of the sphere do these orbitals correspond? How
could each vibration be stimulated? Repeat the exercise for
an forbital (£ = 3). Compare your answers with the represen
tation of the "hydrogen atomic orbitals depicted in Fig. H12
on p. 105, Why should the contour diagrams there resemble
(rather than be identical with) the auxiliary functions you
have drawn? Suppose you took a spherical shell around the
proton in the hydrogen atom and at each point plotted the
value of the wavefunction in the various orbital states, might
you expect the function so obtained to be identical to the
functions you have been drawing in this section? Why is that?
Calculate the latitude of the maximum concentrations of
electron density in the orbitals d , d^, f .
2. Confirm that the functions V , o , Vi.i. M, 3 do indeed
satisfy the differential equation that defines the spherical
harmonics. Use the legendrian operator in Box 11 on p. 124.
Apply the operator £ (see Box 14 on p. 161) to the explicit
expression for the general form of the spherical harmonics
and confirm that Y^ is an eigenf unction of £ ? with eigen
value mti. Prove that the standing wave solutions V^ m ±
Y are eigenvalues of the magnitude of the angular
spin
223
momentum, but correspond to states with zero component of
angular momentum about thezaxis (as should be expected for
standing waves), and are not eigenf unctions of the operator £
(see "quenching).
Further information. See MQM Chapter 3 for a further dis
cussion of the spherical harmonics, their connexion with
angular momentum, and illustration of auxiliary functions. A
pleasing account of the properties of the spherical harmonics
and their relation to states of particular orbital angular
momentum, to waves on a flooded planet, to tidal motion,
and to the vibration of spheres is given by Kauzmann (1957).
See Kauzmann (1957) also for the solution of the differential
equation for the spherical harmonics, and Pauling and Wilson
(1935). Alternative derivations are provided by MQM
Chapters, Rose (1957), Brink and Satcbler (1968), and
Edmonds (1957). The mathematical properties of the spherical
harmonics, and their components the associated Legendre
functions, are given in detail by Abrarnowitz and Stegun
(1965).
spin. The spin is the intrinsic, characteristic, and irremovable
"angular momentum of a particle. A convenient fiction is to
suppose that the spin is the angular momentum arising from
the rotation of a body about its own axis: this model enables
one to recall most of the properties of quantummechanical
spin, and in particular to understand (albeit at only a shallow
level) why charged particles with spin also possess an intrinsic
magnetic moment. The quantummechanical description of the
spin of a particle leads to the following conclusions.
1 . The magnitude of the spin angular momentum of a
particle is determined by the spin quantum number s which has a
positive, unique, integral or halfintegral value characteristic of
the particle. If the value of s is known the magnitude of the
spin can be calculated from the expression \s{s + 1 )] y 'h. As
an example, the spin of an electron is s = \; this means that its
spin angular momentum is 3^fi/2, or 091 X 10" 34 J s, what
ever its other state or condition (the spin is an ineluctable
characteristic of the particle, like its charge). Each nucleus has
a characteristic spin, and the letter / is used in place of s as the
nuclear spin quantum number. See Table 17 for a list of the
spins of some nuclei.
2. The orientation of the spin angular momentum is
quantized (confined to particular angles) in the manner of all
'angular momenta. The orientation is determined by the value
of the magnetic quantum number m : m h is the value of the
component of spin angular momentum on an arbitrary axis in
space (conventionally thezaxis). As an example, two values
of m are permitted for the electron (s = h, namely m =*+«;
corresponding to a component of magnitude ~li on the zaxis,
and m — — ^, corresponding to a component — ^fi on the
zaxis. The different signs are often referred to as denoting an
'upspin' (or aspin) or a 'downspin' (or /3spin), and the good
sense of this can be appreciated from a "vector model of the
situation.
3. Spin is a nonclassical phenomenon in the sense that if
ti were zero the spin angular momentum would vanish. Do not
draw the conclusion that all angular momenta are nonclassical
in the same sense: an orbital angular momentum of magnitude
[£(£ + 1)3 fi could survive the dwindling of li to zero in a
classical world because fi can be increased without limit so that
the product did not vanish; but the value of s is fixed.
4. Spin is not a relativistic phenomenon in the sense
normally put on these words; thus although spin emerges
naturally as a consequence of 'Dirac's relativistic equation it
is possible to arrive at its existence without referring to
relativity.
5. Spin is a fundamental classifier and divides all matter
into two camps with fundamentally different behaviour.
Particles with halfintegral spin are called 'fermions and
satisfy FermiDirac statistics; particles with integral spin
(including zero) are bosons and satisfy BoseEinstein
Statistics. The two classes satisfy different forms of the "Pauli
principle, and because of this they show profoundly different
behaviour. It Is just possible that there exists a third camp
containing the parafermions which are neither fermions nor
bosons and might be needed to account for the properties of
quarks, if these exist.
6. The tendency of spins to •pair is a term too often illused
in fallacious accounts of chemical bonding, where it is quoted
as the reason why bonds form. Energy considerations govern
bonding, and if by pairing electrons are enabled to enter a
lowlying orbital, and so reduce the energy of the molecular
system below that of the separated atoms, then pairing will
224
spincorrelation
occur. But rather than showing any transcendental mutual
affection they are forced to pair (essentially by the "Pauli
exclusion principle) in order to achieve this lowenergy state.
Questions. What is spin? What do the quantum numbers s and
m signify? What angle does the spinmomentum vector make
to the zaxis in the aspin state of an electron? What is the
minimum angular momentum of a photon? Why is spin non
classical? Which of the f ol I owi ng species are ferm ions and wh ich
are bosons: e, p, n, 4 He, 3 He, 2 H + , photon, H 2 , and quark? Why
do electrons seem to show a tendency to pair?
Further information. See MQM Chapter 6 for more infor
mation, especially information about coupling spins together.
See Dirac (1958) for a horse's mouth account of spin, and
Salem and Wigner (1972) for speculations on parafermions. For
the anguiar momentum of spin see "angular momentum and
references therein. For an account of the manifestations of
spin see Wheatley (1970), and McWeeny (1970) for a more
sophisticated version, For an account of other properties that
behave like spin (for example, charge) see Lipkin (1965) and
Lichtenberg (1970). Spin was introduced empirically by
Uhlenbeck and Goudsmit {1925, 1926) in order to explain
features of 'atomic spectra, developed into a consistent theory
by Pauli, and then shown to be a consequence of "Dirac's
equation. The historical development of the idea is described
in §3.4 of Jammer (1966), a book well worth turning to in
order to see the emergence (and sometimes eclipse) of the
unconventional. See also "SternGerlach experiment for an
earlier experiment proving the existence of spin, but not
interpreted then as such. Why spin is not a necessarily
relativistic phenomenon is described by Gaiindo and
Sanchez del Rio (1961).
Spincorrelation. Electrons with parallel "spins tend to stay
apart, and those with opposite spins tend to bunch together.
This remarkable phenomenon has nothing to do with the
charge of the electron (although it affects the average
Coulombic repulsion of two electrons and appears in the
"exchange energy); nor is it, one presumes, supernatural. The
tendency is an intrinsic property of electrons and is a con
sequence of the "Pauli principle. This may be seen by con
sidering a wavef unction for two particles \p{ri, r 2 ), and
supposing that the particles have no interactions. Then the
wavefunction can be written as the product ^ a (r[ li^bta'
where particle 1 occupies orbital a and particle 2 occupies
orbital b. If such a wavefunction is to accord with the Pauli
principle it must be modified to ^ a (ri)i// b (r 2 ) + i// 8 (r 2 ) ]/> b (r ( )
if the spins are paired, and to 4> a (*\)i> b bi) ~ & l (ra)lM lr l) if
they are parallel. In the latter case we can investigate the
probability of finding both electrons at the same point by
FIG. S10. Formation of 8 Fermi
hole. Contours of total wave
function for two noninteracting
particles in a onedimensional
square well. Note that node at
x i = X2 when the spins are
parallel.
Porollel spins
Paired CantiporollelJ spins
spinorbit coupling
225
letting r, and r 2 coincide: the wavefunction becomes
<MM>Mi'i)~!M r ]l  M r i). and vanishes. Therefore we
must conclude that there is a vanishing probability of finding
both electrons at the same point if their spins are parallel.
In Fig. S10 is illustrated the situation for two particles
(electrons, with their charge ignored) in a onedimensional
square well (a narrow wire). The contours depict the prob
abilities of finding the particles at points x, and x 2 in the wire
when one particle occupies the ground state (n = 1, see
•particle in a square well) and the other occupies the first
excited level [n = 2). The mutual avoidance when the spins are
parallel (Fig. S10 a) is manifest as the disappearing probability
along the line x t = x 2 . If one imagines an electron at some
point, then one may imagine a small surrounding volume into
which electrons with the same spin direction will tend not to
penetrate: this region is referred to as a Fermi hole. An
electron with a spin opposite to the first is described by the
other wavefunction, and will have an enhanced probability
of occurring within this small volume; although the resulting
bunching of electrons is not, as far as I know, referred to as a
Fermi heap. Thus we see that the relative positions of electrons
tend to be correlated by virtue of their relative spins.
Further information. See MQM Chapter 8 for a further dis
cussion and another picture. There is a brief account of the
problem in §IV.6 of Linnett (1960), who gives pictures like
Fig. S10 but for the helium atom: these are instructive.
Linnett (1964) has emphasized spincorrelation effects in a
theory of chemical bonding. Reviews of electron correlation in
atoms and molecules include those by Paunch (1969) and
Hylleraas (1964), See "exchange energy and self consistent
fields for more information.
Spinorbit coupling. An electron in an atom possesses a
magnetic moment both by virtue of its "spin and its "orbital
magnetic moments. Two magnetic moments in each other's
vicinity interact, and the strength of the interaction depends
on the magnitudes of the moments and their mutual orien
tation: this interaction energy appears in atomic spectra as
"fine structure.
The source of the spinorbit coupling energy can be visual
ized by taking up a position on an electron which is orbiting a
nucleus of charge Ze. Such an observer will see a positive
current encircling him, and a consequence of the current is a
magnetic moment at hts position. Thus we can conclude that
the electron spin magnetic moment is bathed in a magnetic
field arising from its own orbital motion. (There will also be
magnetic fields arising at the electron by virtue of the orbital
moments of the other electrons in the atom or molecule, but
these spinotherorbit interactions are generally less important.)
It Is in principle a simple matter to calculate the magnetic
field B at the electron by virtue of its orbital motion, and
therefore to deduce its magnetic energy from the expression
"9% S  B (because the interaction energy is — m.B, and the
spin "magnetic moment \sgy^); then we could anticipate that
B is proportional to I, the orbital angular momentum, and so
expect an energy of the form (f/h 2 )s.l, where f is the spin
orbit coupling constant Nevertheless, the calculation is not
quite so straightforward, and the early, straightforward
physicists, who first did it, were perplexed when they obtained
disagreement with experiment.
The calculation proceeds by finding the magnetic field at
a body moving in a electric field with a particular velocity. In
this case the electric field is due to the nuclear Coulomb
potential, and the velocity may be expressed in terms of the
orbital angular momentum through 1 = r A mv. Then the
resulting radiusdependent quantity is averaged over the radial
part of the atomic orbital occupied by the electron, and the
mean constant so obtained is identified with JYh 2 . This
approach gives an answer which is broadly correct (f is
strongly dependent on Z, and varies as Z 4 ) but is wrong by a
factor of 2. With "spin we know that factors of 2 are not
unreasonable; but even so they must be justified. One
justification for this extra factor may be found from the
"Dirac equation; from this the correct expression emerges.
Nevertheless, it is always pleasing to seek a more physical,
visualizable explanation, and as usual one may be provided.
When we step on to the orbiting electron in order to observe
the current due to the central nucleus, we must do so with
more circumspection than had Copernicus when he trod the
opposite journey. In the atom the electron is moving so fast
that it must be regarded relativistically, and watching an
electron spin from the viewpoint of a nucleus is not the same
as watching it spin from the viewpoint of an observer travelling
with it. By a coincidence, or by design, the electron is moving
spinspin coupling
in such a way that to an observer on the nucleus it appears to
be spinning at only onehalf its rate for a travelling observer.
This modification of its motion, which is essentially Thomas
precession, introduces an extra factor ^ into the spinorbit
calculation, and so brings it into conformity with the "Dirac
equation, and with experiment.
The strength of the spinorbit coupling constant increases
with the atomic number of the atom; the heavy atoms have
large spinorbit coupling constants (some are listed in Table 9
on p. 271). In oneelectron atoms this dependence isZ 4 as
mentioned above, and this reflects the dominant sampling by
the electrons of regions close to the nucleus where the field is
strong: as Z increases the orbital contracts and the electric
increases, and both lead to a larger value of J. Outer electrons
are further away, and in hydrogenlike atoms J falls as 1/n 3 :
this also reflects the magnitude of the electric field sampled by
the electron.
Further information. See MQM Chapter 8 for a detailed
examination of the topics mentioned here, including the
calculation of f for hydrogenlike atoms, and a further
mention of Thomas precession. The latter is welldescribed in
Moss (1973) and Hameka (1965). The role of shielding
electrons is examined further in MQM. A thorough discussion
of spinorbit coupling effects, especially with respect to
"singlet and triplet states and "phosphorescence, will be
found in McGIynn, Azumi, and Kinoshita (1969): they
explain how to do calculations involving spinorbit coupling,
and tabulate data. For applications in atomic spectroscopy,
see "fine structure and its ramifications. See Table 9 for some
spinorbit coupling data.
spinSpin coupling. The interaction between nuclei that in
"nuclear magnetic resonance (n.m.r.) gives rise to the splitting
known as fine structure is called spinspin coupling, and its
magnitude is generally denoted J and quoted in hertz {Hz,
cycles per second}. The appearance of a spectrum showing fine
structure is shown in Fig. N3 on p. 154. The coupling between
most protons lies in the range —20 Hz to +40 Hz, although
the commonly observed range is from Hz up to 10 Hz. These
energies are minute, and correspond to a magnetic field being
induced at one nucleus by virtue of the presence and orien
tation of another.
One possible mechanism for the interaction is a direct
"dipoledipole interaction between the two nuclear "spin
"magnetic moments; but the spherical average of such an inter
action is zero, and so it cannot contribute to the fine structure
in a molecule freely rotating in a fluid medium.
Another interaction involves the electrons in the bonds as
intermediaries in transmitting the interaction between the
nuclei. This mechanism may be illustrated by the example of
the hydrogen molecule: the problem is to account for the fact
that the energy of the molecule with the nuclear spins parallel
is different from that of the molecule with the nuclear spins op
posed. The key to the mechanism is the "hyperf ine interaction
between the nuclei and the electrons. Consider the case when
an a electron is close to nucleus A; by virtue of the "Fermi
contact interaction the electron and nuclear magnetic
moments couple, and the lowerenergy configuration is with
antiparallel spins {parallel moments) {Fig. S1 1). But the
"Pauli principle requires the other electron in the bond to be
antiparallel to the first, and charge correlation suggests that
the other electron, with (3 spin, will be predominantly in the
vicinity of the other nucleus. With that nucleus the second
electron has a hyperf ine interaction by virtue of the Fermi
contact term, and so the lowestenergy configuration for the
other nucleus is with 0spin. Therefore, overall we see that if
one nucleus has a particular spin then the other attains the
lowest energy if it has the opposite spin. It follows that the
energy of the molecule with parallel nuclear spins lies very
slightly above that with opposed spins; consequently it
requires energy to turn the spins into a parallel alignment.
FIG. S1 1. Spinspin coupling between protons.
The magnitude of the effect depends on the strength of the
hyperf ine interactions involved, and on the ease with which
the electrons In the bond can be polarized to wash predomi
nantly over the most favourable nuclei: the latter depends on
the mean excitation energy of the bond, and since the
tendency for the electron spins to be decoupled must be
taken into account, it turns out that the mean energy required
is the mean energy of excitation to triplet configurations of
the bond (see "singlet and triplet states). The interaction does
not average to zero as the molecule rotates because the contact
interactions are isotropic and do not themselves average to
zero.
No account of the external magnetic field was taken in the
mechanism, and therefore it should be expected, and is indeed
found, that the spinspin coupling interaction is independent
of the strength of the applied field.
We have shown that the interaction energy depends on the
relative orientation of the two nuclear spins; therefore it is not
unreasonable to expect an interaction proportional to the
scalar product i A .l 8 : this form of the interaction is found by
detailed calculation, and the constant of proportionality is just
the spinspin coupling constant JPh. (If J is positive, the anti
parallel orientation of the spins is the more stable because
W&H A .I B then gives a negative contribution to the total
energy.)
The example we have provided is artificial in one sense: the
spinspin coupling can be detected in an n.m.r. spectrum only
when the coupled nuclei are in different chemical environ
ments (have different "chemical shifts). Thus although there
is a coupling between the protons in H 2 (with a magnitude of
280 Hz), the n.m.r. spectrum consists of a single line because
all the allowed transitions occur at the same energy (see
•n.m.r.). It is more realistic to consider proton coupling in
more complicated molecules, and the mechanism already
described may be extended in a simple fashion by considering
the chain of interactions through the bonds as well as the
possibility of direct interaction by the overlap brought about
by squashing two nonbonded atoms together.
In an obvious notation, using large and small arrows to
denote electron and proton spins respectively, the interaction
in H z can be represented by j t I j \ } for the lower energy
spinspin coupling 227
orientations, and j t I  t  for the upper energy. Analog
ously, in the CH 2 group the chain will be j f I I f [
where the parallel orientation of the middle electron pair is
favoured by the "Hund rule of maximum multiplicity for
atoms (which favours parallel arrangements of spins on atoms);
the upper energy orientation for this group is
\ M i ! i ] J I • and '* ' s i m P ortant to note that,
because of the insertion of the atom in the chain of inter
actions, the parallel nuclear orientation lies below the anti
parallel (corresponding to a negative./). This chain of
interactions can be quite large: J can lie in the range
— 21 5 Hz < J < 424 Hz. An example is the coupling in HCHO
(42'4 Hz). A coupling between two protons separated by three
bonds proceeds through the chain j f \ t \ 1 1 Mt
and leads to an antiparallel arrangement as the lowenergy
state U positive). Beyond three bonds the interaction is
strongly attenuated that is, much reducedthis is fortunate,
for otherwise n.m.r. spectra would be impossibly complicated
to disentangle.
The spinspin coupling between protons separated by bonds
depends on the conformation of the molecule. This is because
the efficiency of alignment of the spins at the intervening
atoms depends on the relative orientation of the bonds. Thus
the predominance of —{it ■■■■{ over {•••♦fii—*}
depends on factors like the hybridization of the atoms. As an
example, the trans coupling in ethene is 191 Hz whereas the
ci's coupling is 116 Hz.
Although we have dismissed the direct dipoledipole inter
action in fluids, the electronnucleus dipolar hyperf ine
interaction can contribute to spinspin coupling. We have
seen that the spintransmission mechanism involves two
hyperf ine interactions, one at each end of the chain; the
rotational average of the product of two dipolar interactions
does not disappear, and so two electronnuclear dipole inter
actions, one at each proton, can contribute. This type of
interaction is important in atoms other than hydrogen where
porbitals occur in the valence shell.
Questions. What is the significance of the term 'spinspin
coupling' in n.m.r.? Under what circumstances will a
Stark effect
coupling not show in the spectrum even though it is nonzero?
What is the range of proton coupling constants in n.m.r.?
What dependence on the strength of the applied field do they
show? Why is the direct dipoledipole interaction between two
spins unimportant in fluid media? Is it important in solids?
What is a typical magnitude for the direct dipolar interaction
between the two protons in the water molecule? Investigate
how this interaction depends on the orientation of the mole
cule with respect to the applied field {assume that the proton
moments align themselves with respect to this field and search
Box 5, p, 50, for formulae). What structural information
might you anticipate obtaining from a study of the direct
interaction? What is the source of the spinspin interaction
between protons in liquids? Explain the sequence of inter
actions that transmits the orientation of the proton to its
neighbour. What happens to the interaction when the protons
are separated by 3, 4, and 5 bonds? What excited states should
be mixed into the ground state in order to yield the spin
polarization of the bond? Can a dipolar interaction give any
contribution in fluids? What is the significance of a negative
value of J?
Further information. See MQM Chapter 11 for a detailed dis
cussion of the source of the spinspin coupling. The role it
plays in n.m.r. is described by McLauchlan in Magnetic
resonance (OCS 1). Further details will be found in Lynden
Bell and Harris (1169), Carrington and Mctachlan (1967),
Slichter (1963), Memory (1968), and Abragam (1961). The
analysis of n.m.r, spectra in order to extract coupling data,
and its application, are described by these books and in
Roberts (1961), Abraham (1971), Emstey, Feeney, and
Sutdiffe (1965), Pople, Schneider, and Bernstein (1959), and
Corio (1966), The reason why the spectrum exhibits coupling
between nonequivaient nuclei can be seen by referring to the
little piece of mathematics in §4.4 of Carrington and
McLachlan (1967). Tables of J values will be found in Emsley,
Feeney, and Sutdiffe (1965).
Stark effect. The modification of the energy levels, and
therefore of the spectra, of atoms and molecules by the appli
cation of an electric field, is known as the Stark effect. It is
possible to distinguish the first and secondorder effects
(which are respectively linear and quadratic in the strength of
the applied field) and the atomic and molecular effects.
The firstorder atomic Stark effect is large but rare, for it
depends on the presence of a degeneracy which enables the
atom to respond massively to the applied field. Such a
situation occurs in atomic hydrogen: because the 2s and 2p
orbitals are degenerate, when a field is applied the electron
can easily reorganize itself by hybridization. Thus the com
bination tp 2s + i*%p S' ves a distribution strongly biased
towards the lowpotential region, and the other possible
combination i/^s — lK p is concentrated on the highpotential
side of the nucleus (Fig. S 12). Therefore the transitions
involving the n = 2 shell of the atom arc no longer degenerate
and occur at different frequencies. The p  and p orbitals
x y
are unaffected by the field.
field off
field on
FIG. S12. Linear Stark effect in atomic hydrogen.
As in the Zeeman effect the transitions are polarized. Two
points follow from our discussion of electric dipole
transitions.
1. If m% does not change, the emitted light (forming the
irlines) is polarized parallel to the direction of the applied
field (and so it would not be seen if viewed along the direction
Stark effect
229
of the field; it is radiated in a belt around the transverse
direction).
2. If m e changes by +1 the light (the allnes) is transversely
polarized. When viewed perpendicular to the field the
Amj; = +1 component is right circularly polarized and the — 1
component is left circularly polarized. When viewed along
the field direction the light is unpolarized because the +1 and
—1 transitions occur at the same frequency, and so the light
emitted from different atoms gives an incoherent superposition
of both circular polarization states.
Not only does the Stark effect cause a splitting of lines, it
also causes them to broaden slightly, and shifts the series
limits to lower frequency. Both effects are related to the presence
of the low potential on one side of the atom, for it enables an
electron to escape (Fig. S13). An electron in a state not far
from the ionization limit may be able to tunnel through the
remaining potential energy barrier and emerge into a region
where the applied field can pluck it from the atom. This
reduces its lifetime in the excited state, and so, by the
"uncertainty principle, the energy of that state is made
imprecise; this imprecision appears in the spectrum as a
broadening of the appropriate lines. The electron need not
be excited into so high an energy state for ionization to occur,
and so the field also reduces the energy of the series limit
(Fig. S13).
appoint J.R
FIG. S13. Extra consequences of the Stark effect for hydrogen.
In atoms not having the degeneracies possessed by hydrogen,
the firstorder effect is absent and is replaced by the much
weaker secondorder effect : One power of the field is used
in distorting the atom from spherical symmetry ("polarizing it),
and the second power is used in the interaction with the dipole
moment of the distorted atom, so causing the energy
separation. Since normal fields can polarize atoms only slightly
the induced dipole is small and its interaction with the field
weak; therefore the secondorder Stark energy shifts are small
and huge fields (~ 10 s V cm" 1 ) must be used.
The molecular Stark effect may also be of first or second
order. The firstorder effect is shown by symmetric top
molecules with permanent dipole moments: the applied field
causes "rotational states of the same value of J but different
values of M to have different energies (they are degenerate in
the absence of the field), and the splitting is proportional to
the permanent dipole moment of the molecule (i, as we
explain below. Since fields of the order of 50 kV cm" 1 give
splittings of the order of 20 MHz, and this is easily and
accurately detectable in a microwave spectrum, the method is
a powerful way of determining dipole moments.
It is instructive and quite easy to understand why the
energy of a state with quantum numbers J, K, and M is
shifted by an amount pMKBUKJ + 1 ) by a field E , From the
theory of the "rotation of symmetric top molecules we know
that K is the projection of the angular momentum J on the
figure axis, and that J "precesses around this axis. An alterna
tive view would be to consider the vector J as fixed and the
axis as precessing around it. Since the dipole is directed along
the axis the motion averages out its components except for
one of magnitude jncosO along J (see Fig. SI 4), But J is also
precessing about the field direction, and the component of
dipole parallel to the field isjucosfl cost 1 '; therefore the inter
action energy is — jufcosfl cosf?'. From Fig. SI 4 we can use
simple trigonometry to express cost 1 as Kl[J(J + 1)) % and
cos8' as M/UV + 1 )] Vl . Combining these results we obtain
the energy as —yMKEUV + 1 ), as we anticipated. The appli
cation of fiie "selection rules &J= ±1, AK = 0, and AAf = 0,
±1 enables the spectrum (and from the M changes, the
polarization of the lines) to be predicted.
In the case of linear molecules, where the angular
momentum is perpendicular to the dipole moment, and in
230
SternGerlach experiment
FIG. S14. Molecular Stark effect.
asymmetric tops, where the lack of symmetry causes the
motion to be very complicated, there is no linear effect, but
the orientating effect of the field (which, in classical terms,
distorts the rotational path of the molecule) affects the
energy of the states and induces energy shifts of the form
IJ 1 E 2 {A 1 + A t M 2 ).
The molecular Stark effect is of considerable importance in
the study of permanent dipole moments of the molecules that
can be examined by microwave spectroscopy, and it is also
important in the technology of microwave spectrometers, for
an oscillating electric field (usually of several 10 kV cm" 1 ) will
modulate the position of spectral lines, and therefore the
intensity of absorption or emission at a particular frequency.
Detectors making use of this oscillation of the intensity are
called Starkmodulation spectrometers.
Questions. 1 . What is the Stark effect? What classifications of
the effect are there? Why is the firstorder atomic effect con
fined to hydrogenlike atoms? Would you expect it to be
shown by helium in a highly excited state? What polarizations
are observed for the permitted transitions in hydrogen? Take
the hydrogen energylevel diagram (Fig. G3 on p. 86), modify
it to show the effect of an electric field, and discuss the form
of the spectrum. The shift of the energy of an electron in the
n = 2 shell results in a separation of the levels by an amount
Qea a E, where a is the "Bohr radius. Compute the separation
for an applied field of 20 kV cm" 1 and 100 kV cm' 1 . What
separation would be expected for He + in the same state and in
the same fields? What other effect does a strong field have?
What effect has the field in the case of atoms that lack
hydrogenlike degeneracy? Why is the secondorder effect
small? What effects occur in molecules? The permanent dipole
moment of ammonia is T47 D; discuss the splitting of the
J = 0, 1 , 2 states brought about by a field of 50 kV cm" 1 ;
sketch the form of the spectrum and label the polarizations of
the lines (ignore, if you are aware of them, the effects of
•nuclear statistics).
2. Using the explicit orbitais given in Table 15 on p. 275 set up
the 2 X 2 secular equation for the effect of an electric field
along z on the energies of the 2s and 2porbitals of hydrogen.
Solve for the energies and states, and evaluate all integrals.
The splitting of the 2s and 2porbitals is, as stated in the
proceeding question, 6ea E, and this should be your answer.
Further information. See MQM Chapter 8. A nice, and
occasionally anthropomorphic, discussion of the effect on
atomic spectra is given in Chapter 17 of Condon and Shortley
(1963); see also §111 A4 of Kuhn (1962), Chapter 20 of
White (1934), and §11.3 of Herzberg (1944). The molecular
effect, together with accounts of how to determine dipole
moments and build spectrometers using Stark modulation, is
described in Chapter 7 of Sugden and Kenney (1965) and in
Chapter 10 of Townes and Schawlow (1955).
SternGerlach experiment. The SternGerlach experiment
consisted of the passage of a collimated beam of atoms (silver
atoms boiled off hot metal through slits into a vacuum)
through an inhomogeneous field, and the observation of the
distribution of the atoms when the beam had been deposited
on a glass plate. If the atoms possessed a magnetic moment
the effect of the inhomogeneous field would be to drive in one
direction those that had one orientation, and in the opposite
direction those that had an opposite orientation, and, accord
ing to classical physics, to spread atoms with all intermediate
orientations into the region in between. An inhomogeneous
magnetic field is essential for this effect because a homogen
eous field would not split the beam as it provides no
directional information; an inhomogeneous field, one with a
nonvanishing gradient, provides a sense of direction.
In their first experiment Stern and Gerlach observed that
classically anticipated result. In their second, done with great
care with a low pressure and a long exposure, they saw that
the band of deposited atoms had two closely spaced com
ponents, separated by a clear region. This result is wholly at
variance with classical physics but in full accord with
quantum theory, for each silver atom possesses an unpaired
electron with "spin, and therefore has a 'magnetic moment.
Quantum theory predicts that a spinj object can take only
two orientations in a magnetic field, and so the Stern
Gerlach experiment confirms this in a striking fashion.
It is important to note that this was the first piece of
evidence for the quantum theory that did not involve a
thermal experiment or an experiment involving radiation: it
provided a purely mechanical demonstration of quantization
{space quantization, as the restricted number of orientations
of an angular momentum is termed). The original explanation
did not associate the magnetic moment with the intrinsic spin
of the electron; that came later (1925) when Uhlenbeck and
Goudsmit introduced the concept from their study of atomic
spectra. The experiment was also one of the first applications
of molecular beams, a subject now in a dynamic phase.
Questions. Sketch the StemGerlach experimental arrangement.
Why is it necessary to use a very low pressure in the apparatus?
Why is it necessary to use an inhomogeneous field? What is
the result predicted by ctassical physics for the experiment?
What is the result obtained? Why is the result consistent with
quantum theory? Suppose the upper beam were passed through
another inhomogeneous field with (a) the field in the same
direction as the first, and (b) the field rotated about the
direction of the beam by 90°: what would be the result
of the experiment?
Further information. The SternGerlach experiment is put
into its historical context in §3.4 of Jammer (1966). For the
details, see §134 of Richtmeyr, Kennard, and Lauristen (1955)
and §V1,2 of Ramsey U956). The original papers are by Stern
(1921) and Stern and Gerlach (1922). An analysis wilt be
found in §14.16 of Bohm (1951). For the philosophical dis
superposition principle 231
cussion to which the result gave rise, see Einstein and
Ehrenfest (1922) and Jammer (1966). The modern state of
molecular beams is described in Ross (1966) and Ramsey
(1956); see also Levine and Bernstein (1974).
superposition principle. The superposition principle states
that when a situation is a composition of a number of
elementary situations, its amplitude is the superposition of the
amplitudes for the components. The significance of this
principle, which is one of the fundamental principles of
quantum mechanics and implies the notable features of the
differences between classical mechanics and quantum
mechanics, can be introduced by considering the classical
situation.
Suppose that an event can be divided into a number of
composite events; for example, the event might be the journey
from a point p x to another p 2 , and the elementary events
might be the journeys by alternative paths through the points
P a or P b  Classical theory ascribes a probability P{a) to the
path through p a and a probability P(b) to the path through p ,
and goes on to say that the total probability of making the
journey from p, to p 2 is the sum of the probabilities of making
the individual journeys: P = P(a) + P(b) (Fig. SI 5).
Classical
PCa)
PCM
(/ —+0
FIG. S15. The superposition principle.
PCoHPCW
Quanta!
Pidi=f\o)fto)
pca=Y*tw^cb)
l^faj+^cal"
232
symmetry operation
This discussion might seem trivial; but quantum theory
shows that it is wrong. The superposition principle agrees that
there is a probability of effecting the journey through p with
a probability P{a), and an analogous probability for the path
through p b , but it disputes the assertion that the joint prob
ability is the sum of the probabilities. It implies that in order
to compute the joint probability it is first necessary to find
the probability amplitude (the °wavef unction ) for the
individual paths, then to find the probability amplitude for
the composite situation, and from that to find the probability
itself (Fig. S15). Let the probability amplitude for the journey
through p be i^(a): by this we mean that the probability itself
is ^*(a}i^{a). The corresponding probability amplitude for the
journey through p is !^(b). The superposition principle tells
us that the amplitude \p for the composite process is the sum
<^(a) + \jj{b). It follows that the probability for the joint
process is $*$), which may be expanded into P(a) + P(b) +
^*{a)^ib) + i^(a)^*{b}. This total probability differs from
the sum of the individual probabilities by the interference
term ip*{a)ip(b) + 0{a)i//*<b).
First we should note that if one of the paths were forbidden
its amplitude would be zero and the probability for the overall
journey would reduce to P = P(a) (for p b closed). Next we
should note that the interference term does not occur in the
classical case, is characteristic of quantum processes, and arises
from the insistence of the superposition principle that ail basic
manipulations are done on the amplitudes.
If the superposition principle is a true description of
Nature we should be able to detect some direct physical con
sequences: one that might immediately spring to mind is the
possibility of observing interference effects for particles that
are able to travel from a gun at p t to a target at p 2 through a
screen perforated with two holes: such interference effects of
particles, which correspond to the analogous interference of
light under similar circumstances, have been observed.
The superposition principle applies to all the processes of
quantum mechanics. Wherever a compound situation is under
consideration the calculations must be performed on the wave
function (the state amplitude): that is why the "Schrodinger
equation is a central formula of quantum theory— it enables
the amplitude of a state to be calculated, and shows how to
determine its time evolution.
The LCAO method is an example of the working of the
superposition principle. To see this, we can take the view that
the probability of finding an electron at a particular point of
the molecule depends on its probability of being in some of
the atomic orbitals of the constituent atoms. Therefore to work
out the probability of being at the point r we must know the prob
ability amplitude for the electron to be there if it occupied
each of the atoms separately. These probability amplitudes
are just the atomicorbital wavef unctions themselves, each
evaluated at the point r. Therefore the total probability
amplitude of the electron being at r is a sum of the atomic
orbitals (at that point), and the probability is the square of
this amplitude. This is precisely the interpretation of 'mole
cular orbitals in terms of the LCAO approximation.
Questions. State the superposition principle. Discuss the
process of travelling from Aix to Ghent through two gated
roads, and take into account both classically and quantum
mechanically the chance that the gates might be closed.
Discuss the analogy between the propagation of particles
and of light in terms of the superposition principle. Discuss
the diffraction of particles in terms of the superposition
principle. Why is the LCAO account of molecular structure
plausible in terms of the superposition principle? Discuss,
in its terms, the concept of "hybridization.
Further information. See Feynman, Leighton, and Sands
(1963), §9.2 of Bohm (1951 ], and Chapter 1 of Dirac (1958).
. For a simple account of optical analogies see MQM Chapter 2.
See Jammer (1966) for historical attitudes. See Feynman and
Hibbs (1965) for a construction of quantum mechanics in
terms of the superposition principle.
symmetry operation. A symmetry operation or symmetry
transformation is a transformation of the coordinates of the
system {passive convention) or transformation of the object
itself {active convention) that, after its application, leaves the
system in a configuration indistinguishable from its original.
The active convention is the more direct, and we illustrate
it first. Consider an undecorated square object lying on a plane
on which is drawn a coordinate system; close your eyes, rotate
the object through 90° in the plane, and look at it again. It is
impossible to tell that it has been rotated, and so this operation
symmetry operation
233
FIG. SI 6. (a) Active and (b)
passive conventions for
transformations.
is a symmetry operation. Had the rotation been through an
angle less than 90° we would have noticed a new orientation.
Similarly a rectangular (oblong) object can be rotated through
180 into an indistinguishable position, which rotation is
therefore a symmetry operation; but rotation through 90° is
not a symmetry operation (Fig. S16),
In the passive convention a coordinate system is drawn on
the plane on which the square object is lying; if we move a
units along x we arrive at the edge of the square (of side 2a),
and if we step a units along y we also come to an edge. Now
rotate the coordinates (that is, the underlying plane rather
than the object) until thexaxis lies along the direction
FIG. S17. The symmetry
operations C , o , i, and S and
an example of a molecule
possessing each.
234
sy m met ry ope ration
originally occupied by the yaxis. Stepping along the new
xaxis a units, we again encounter an edge; likewise for the
new yaxis. This system is indistinguishable from the first, and
so the 90 rotation is a symmetry operation. In the case of a
rectangle of sides 2a and 2b stepping along the original xaxis
a units will bring us to an edge, but if we step a units along the
rotated xaxis we shall not be at an edge unless a = b (Fig.
SI 6). This is therefore not a symmetric situation, and so a 90°
rotation of a rectangle is not a symmetry operation.
It is a matter of taste which convention one employs,
although there are aesthetic advantages (at least to my muse}
in the passive convention on the grounds that it is always
possible to manipulate coordinates (they being a convenient
mathematical fiction, and at our disposal), whereas it may not
be possible to do the active act. This is especially true in the
case of symmetry operations other than rotations: inversion
through a point is difficult to effect actively (although it may,
of course, be imagined), and so too is reflection.
The basic symmetry operations are those of translation
(especially in periodic systems such as crystals or in free space),
rotation through 2ir/n rad (denoted C ), reflection in a plane
(a)— a subscript v implying a vertical plane and an/i implying
a horizontal plane— inversion through a point (i), and rotary
reflection (S n ), which is a composite motion effected by
rotating by 2itln rad and then reflecting in the plane to which
the rotation axis is perpendicular (see Fig. S17).
The study of the effect of symmetry operations is the
basis of "group theory.
Questions. 1 . What is a symmetry operation? What two con
ventions are employed? Discuss in the active and the passive
conventions the operations denoted C 2 , C 3 , (J, i, S*. Enumerate
the symmetry operations of the following objects: a left
hand, a right hand, a man, a chair, a cube, a triangle, a
tetrahedron, a square pyramid, the molecules H 2 0, C0 2 , CH 4 ,
SF 6 , benzene.
2. Demonstrate that an alternative and consistent definition of
a symmetry operation is that it is an "operation that
■^commutes with the system's "hamiltonian.
Further information. See MQM Chapter 5 for a further dis
cussion of symmetry operations and the development of the
structure of "group theory. See Atkins, Child, and Phillips
(1970) for a simple enumeration of symmetry elements and
some examples. Refer to the section on "group theory for
further references, and see Weyl's book (1952) for an enter
taining pictorial account of the appearance of symmetry in
Nature.
T
tensor. To a mathematician a tensor is an object that trans
forms in a particular way when the coordinates he is using to
describe a problem are transformed. A scalar quantity, which
may be regarded as a zerothrank tensor, does not change
when the coordinates are transformed. A vector does change
when the coordinates change (at least, the 'object' remains
the same, but its description in terms of the new axes is
different from its description in terms of the original axes) :
it is a firstrank tensor. More complex objects may be given
as examples of higherrank tensors; for example, the object
rr, where r is a vector, is a secondrank tensor. The mathema
tician distinguishes between contravariant and covariant
tensors: for example, the set of coordinates [x, y, z] of a
vector transforms as a contravariant tensor of rank one, and
the set of objects [df/ax, bfloy, bflbz] , where r"is a scalar
function, transforms as a covariant tensor of rank one. Tensor
calculus in its most powerful form is concerned with general
transformations of coordinates, and is a powerful tool when
noneuclidean geometry has to be studied: for instance, in
general relativity. When the transformations are limited to
orthogonal transformations, those preserving angles in the
transformation (like a simple rotation of axes), the proper
ties that an object must possess to be a tensor are less
stringent, and the objects that comply are termed cartesian
tensors. When the objects transform in a special way under
rotations we encounter the irreducible spherical tensors;
these are of considerable importance in the discussion of
molecular and atomic properties.
To a chemist a tensor is a quantity that expresses the
directional dependence of the properties of molecules and
solids; he is normally on safe ground because usually, but not
always, the properties that interest him transform in the
same way as tensors, and so are tensors. As an example
consider the effect of an electric field on a molecule. Let the
field be along some axis Z, then the field "polarizes the
molecule and induces a dipole moment. Usually the major
component of this induced dipole lies in the same direction
as the field, but in general the field and the induced dipole
are not exactly colli near, and there are also X and V
components (Fig. T1 ). The magnitudes of the components
are proportional to the strength of the applied electric
field E z , and so the dipole has components a E ,
a yz^z' anc ' a zz^z' wnere tne a QQ are tne coefficients of
proportionality, the "polarizability. If the field is applied
FIG. T1. The effect of offdiagonal elements of the polarizability
tensor. When E is along an arbitrary direction it may induce a dipole
which need not be parallel to E. The dipole is parallel if E lies along a
principal axis (one of the axes marked on the molecule).
235
236
terms
along the Xor Vaxes similar expressions could be set up
to describe the induced dipole (which may have different
components if the molecule is anisotropic}. In all we g et /
nine quantities of the form « QQ ,. Recall that the quantity
rr, with the nine components XX, XY, XZ, . . . ZZ is a
secondrank tensor; then we see that by analogy we may call
the nine quantities a QQ , the components of a secondrank
tensor a, the polarizability tensor. Knowing a we may predict
the polarization in any direction when a field is applied in
any direction.
Other physical quantities that behave like tensors include
those representing a strain a stress type of relation, and it is
from this that the name tensor arose. In the example just
described the stress is supplied by the applied field, and the
strain is the resulting polarization. The magnetic suscep
tibility is of the same type, and may be expressed as a second
rank tensor. The elasticity of a body is of this form, and it
too may be expressed tensoriaily. For rather special reasons
the jtensor of "electron spin resonance is not a true tensor,
and therefore we have called it the ^value; this is a slight
pedantry because most people do call it a tensor, which in
many cases it very nearly is.
Aspinor is a quantity that resembles a tensor, but has the
peculiar property that when it is rotated through 360° it
changes sign; therefore a rotation of 4rr must be made to
bring it back to its initial form. Objects such as this are in
volved in the mathematical description of spin; hence the
name.
Further information . An excellent and moderately simple
introduction to the general mathematical theory of tensors
will be found in Synge and Sen i Id (1949); see also Kyrala
(1967) and Jeffreys (1931). The role that tensors play in the
discussion of the physical properties of matter is described
by Brink and Satchler (1968), Rose (1957), and Edmonds
(1957). See Fano and Racah (1959) for a consolidated
account. Why g is not a tensor is described in §15.6 of
Abragam and Bleaney (1970).
terms. The word 'term' first entered spectroscopy with its
colloquial meaning, and therefore without much fundamental
significance. In the early days of atomic spectroscopy it was
found that a" considerable simplification of the description of
the spectrum of an element could be obtained by expressing
the frequency of every line as the difference of two terms:
since one term contributed to a number of spectral lines it was
necessary to list far fewer term energies than transition
energies. The rule that the spectrum could be so expressed is
known as the Ritz combination principle (terms are combined,
by subtraction, to give the frequency or energy of a transition).
Thus initially the 'terms' of a spectrum were just a collection
of numbers which could be used to predict spectra; but, rather
oddly, it was found that not all combinations (7"j  7~ 2 )
corresponded to an observable transition.
The quantum theory provided a natural interpretation of
terms: it revealed that a term was an energy state of the atom,
and the combination principle was elucidated partly by "Bohr's
postulate and later by proper quantummechanical calculation
which showed that a spectral line represented the transition of
the atom from one stationary energy state to another, and that
the frequency was determined by hv=T\ — T<i. Quantum
mechanics also permitted the calculation of "selection rules,
which tell whether a given combination of terms is observable.
A term is now interpreted as an energy state of an atom, and
a term symbol, which labels the energy state, may be con
structed from the quantum numbers that define the energy
state. A typical term symbol looks like 2 S or 3 P (but other
indices are often attached for a finer description of the state,
as we shall describe). The symbol, illustrated in its full glory
in Box 26, is constructed as follows.
BOX 26: Term symbol
multiplicity
value of I —
(2S+1) M
— state
level
L = 0, 1 , 2, 3, 4, .
X = S, P, D, F,G,
1. The letter denotes the value of L, the total "orbital angular
momentum quantum number, according to the correspondence
L = 0, 1, % 3,4, . . .++ S, P, D, F, G
2. The left superscript is the value of 2S + 1 , where this S is
torsional barriers
237
the total 'spin angular momentum quantum number for the
atom {a massive intellect is not required in order to avoid
confusion of the S with the S of L = 0), The value of 2S + 1
is often referred to as the 'multiplicity of the term, and the
propriety and significance of this are described under that
entry. Thus 2 S is read as a 'doublet S term', and 3 P as a
'triplet P term'.
3. The 'levels of a particular term are distinguished by a right
subscript of the value of J; for example, a doublet term with
L = 1 (that is, 2 P) has two levels with J= jor §: the levels are
denoted 2 P l/2 or 2 P Vi , respectively, and read 'doublet P
onehalf or 'doublet P threehalves'.
4. The state of an atom can be expressed in even greater
detail if one also quotes the value of M Jt the projection of J
on some axis: this label is added as a right superscript. For
example the Mj = — 1 state of the J = 2 level of a triplet
term with L = 2 would be labelled 3 D 2 ~' , and from this
symbol we could write the values of the quantum numbers
5. L,J, and/W,[seeBox26).
A list of terms and their energies enables the spectrum to
be reconstructed; but to do so it is also necessary to know
the permitted combinations: these are determined by the
"selection rules which are normally expressed as the values
of AS, AL , AJ, and AMj permitted in a transition. Inspection
of the term symbol in conjunction with these rules enables
one to see very clearly which transitions are allowed.
Term symbols are also encountered in molecular spectroscopy,
but as L is then not a good quantum number the labels usually
denote some "grouptheoretical classification.
Further information. See MQM Chapter 8 for a discussion of
atomic spectra and Chapter 10 for molecular spectra: both
Chapters illustrate the use of term symbols. Further infor
mation will be found under "atomic spectra, "selection rules,
"fine structure, and "multiplicity.
torsional barriers. The classic example of a barrier to free
rotation is that in ethane: one methyl group cannot rotate freely
about the CH; bond because of its interaction with the other
methyl group. The lowestenergy conformation is the
staggered arrangement of the two methyls, where a view along
the C— C bond shows that the C— H bonds of one group bisect
harmonic
Ceo sine)
potential
FIG. T2. The torsional barrier in ethane, and the fit of a harmonic
potential for small librations.
deep
oscillator
ocm
FIG. T3. Energy levels of free rotor (on the )eft> are modified by a
periodic potential and in the limit of a deep potential become those of
two independent oscillators. Note how the degeneracies change, q 
, wtwe V is the depth of the well.
iV/211 1 ,
238
torsional barriers
the H— C— H angles of the other; the highestenergy confor
mation is the eclipsed, when the C— H bonds of each group
are in opposition. If one methyl group is twisted through a
smalt angle from the staggered conformation, and then
released, it will execute torsional oscillations. As more energy
is added to this twisting mode the oscillations get larger, and
soon the methyl group is able to jerk from one energy
minimum to another. When sufficient energy has been added
the group rotates in a manner that is almost indistinguishable
from a free rotation. Thus a torsional vibration becomes a
free rotation. Conversely, a free rotation becomes localized as
a torsional oscillation as an energetic molecule loses energy.
The quantummechanical explanation of the process treats
the system as a particle in a potential which varies as cos(30/2)
{see Fig, T2): this has the correct periodicity of the actual
system but is only a moderately good approximation to the
actual dependence on the torsional angle 8 (which in detail
is very complicated). At the troughs of the cosine function the
potential vaies 3S 6 2 (Fig, T2), and so a particle confined close
to the minima behaves like a "harmonic oscillator. It follows
that the ground state of the ethane torsion is a zeropoint
rocking about the bond. The first excited state of the torsional
mode resembles a harmonic oscillator in each of the wells; but
the correspondence is exact only for infinitely deep potentials.
In the real molecule the methyl group can "tunnel from one
well to another, and so the wavef unctions are not truly those
of a harmonic oscillator.
At high excitations, when the total energy greatly exceeds
the peaks of the barrier potential, the potential can be ignored.
In this limit the group behaves like a particle of mass 3m
confined to a ring.
The mathematics of the situation can be illustrated nicely
if we turn from the methyl group to an example where there
are only two wells, not three (Fig. T3). For deep wells every
energy level is doubly "degenerate because the rocking group
is in either of the two equivalent wells; for vanishingly shallow
wells every level except the lowest is doubly degenerate because
a free rotor may rotate clockwise or anticlockwise with the
same energy. (The lowest level, corresponding to a static rotor,
is nondegenerate. ) As the barrier is strengthened the latter
situation must pass over into the former. This may be demon
strated explicitly, because the Sch rod inger equation with a
harmonic potential is a Mathieu equation, and its solutions
are tabulated. In Fig, T3 we show the energies and in Fig. T4
FIG. T4. Torsional wavefunctions for shallow and daep walls: note how the functions are more those of a rotor in the former, and. independent
oscillators in the latter.
transition probability
239
the wavefunctions for barriers of different heights. These
pictures illustrate the preceding discussion, for the emergence
of the freerotor energies and wavefunctions is clearly apparent.
Further information , Helpful discussions of the hindered
rotation problem are given in §8.3 of Sugden and Kenney
(1965) and §12.6 of Townes and Schawlow U 955). Solutions
and properties of the Mathieu equation are given in §20 of
Abramowitz and Stegun (1965), and they give numerical data.
See also "inversion doubling for a related situation. A review
of the internalrotation problem has been given by Strauss
(1968).
transition probability. A transition probability is the prob
ability that a system will change from one state to another. Its
principal significance is that it determines the intensity of
spectral lines and so, for poets, the brilliance of a colour. The
quantum mechanical discussion of transition probabilities is
normally based on perturbation theory, for the application to
the molecule of an oscillating electromagnetic field distorts it
slightly, and this distortion can be interpreted as an admixture
of excited states into the original state. Since the *wavefunction
now contains a component of the excited states there is a
nonvanishing transition probability.
The transition probability depends on the strength of the
light (the energy of the perturbation), on the match between
the frequencies of the exciting radiation and the excited
transition (for the coupling is strongest when there is
'resonance), and on the strength of coupling between the
molecules and the electromagnetic field. The last is related to
the transition dipole (see "electric dipole transition), which is
essentially a measure of the extent of movement of the charge
during the transition: if the motion is great, then the field can
interact strongly, and the spectral line is intense.
As an example, consider the excitation of that wellworn
species the hydrogen atom. The 2p ; *— 1s excitation occurs
with a movement of charge from a spherical distribution
about the atom to one clustered around the zaxis; the
transition can be pictured as an oscillation of charge backwards
and forwards along the zaxis, and the dipote moment associ
ated with this motion ensures that the line appears brightly in
the spectrum. The 2s «— Is transition, on the other hand,
although it involves a significant migration of charge, does so
with a conservation of its initial spherical symmetry, and so
there is no dipole moment associated with the oscillation from
one state to the other; the light field does not interact, and the
transition does not occur in the spectrum.
BOX 27: Transition probability
Fermi's Golden Rule
Transition rate (rate of change of probability) for a
migration from a state / to a state f under the influence
of a perturbation of strength V is
where V fj = {\p f \V\\jj.} and p{v f .) is the energy density of
incident radiation at the transition frequency V...
Electric dipole transitions
B fr \d f A 2 /6e tf = B ff .
8 is 'Einstein's coefficient of stimulated absorption or
emission. The total emission rate is
A ff =Bi!h(v fj lc} 3 B fj in general
= (8TrV3eoC 3 h)i£.ld ft .P for electric dipoles.
Transition dipole
d f . = fdr\j/*d \p. d = er.
Fermi's Golden Rule is the mathematical expression for the
determination of a transition probability. It is set out in Box 27,
and the form quoted may be interpreted as the product of the
square of the transition dipole moment between the states (the
dipole acts as a handle for the modification of the amplitude
of the wavefunction, and to get the probability the square must
be taken) multiplied by the energy density of radiation at the
transition frequency (the more powerful the stimulus the greater
the transition probability). The formula may be deduced from
timedependent perturbation theory.
240
tunnelling
This account of transition probabilities underlies the dis
cussion of "selection rules, which enable one to predict when
a transition probability is nonzero. The strength of the
transition dipole is often quoted in terms of the "oscillator
strength, or in terms of the Einstein A and 8 coefficients.
These are conveniently related to the "extinction coefficient,
which is an experimental measure of the spectral intensity.
Questions. 1 . What does the transition probability determine?
Why does the application of a timedependent perturbation
induce spectral transitions? At what frequency is the absorption
strongest? What is a transition dipole moment? Is there a non
vanishing transition dipole moment associated with the
following transitions: 2p y « — 1s, 3p v * — 1s, 3d„„ « — 2p„
2p.
r x "' x ' xy x
2p , 3p * — 2p ? Consider the transition moments
associated with the emissions 2p x — * 1s and 2p^ — > Is: would
you expect the polarization of the emitted light to be different
in the two cases?
2. The xcomponent of the transition dipole moment d* f . for
the transition from a state described by a function \j/ / to one
described by a function \jj f is equal to the integral ~6JdT\pfX\jJ.,
with similar expressions for the y and zcomponents. The
transition probability is proportional to Id^i*. Show that the
stimulated absorption intensity for a particular transition is
equal to trie stimulated emission intensity. Calculate the trans
ition dipole for the transition from the ground state of a particle
in a onedimensional square well to the first excited state
(use the information in Box 15 on p. 166). From the infor
mation in Tables 11, 14, and 15 calculate the intensity of trans
itions from the ground state of a onedimensional harmonic
oscillator to the first and second excited states, and for the
transition 2p < — 1s of the hydrogen atom.
Further information. See MQM Chapters 7 and 11. Transition
probabilities are described by Eyring, Walter, and Kimball
(1944), Heitler [1954), Kauzmann (1957), and the standard
works on quantum theory; see Davydov (1965), Schiff (1968),
and Messiah (1961). Transition probabilities occur in the
discussion of "selection rules, "oscillator strengths, "Einstein
A and 8 coefficients, "polarizabilities, and "intermolecular
forces. A nice, simple discussion will be found in Loudon
(1973).
tunnelling. Quantummechanical tunnelling is the process
whereby particles penetrate potential barriers and appear in
regions forbidden to them in classical mechanics. Consider a
■particle trapped in a square well by a potential barrier of
finite height and width. Let the particle have an energy which,
according to classical mechanics, is insufficient to carry it over
the potential barrier. The quantum mechanics of the situation
shows that the particle has some probability of tunnelling
Through the barrier and escaping from the well. The tunnelling
probability arises from the requirement of the continuity of
the wavef unction at the walls of the well: if the wavefunction
has a nonzero amplitude at the inside edge of the barrier
{and this is permitted when the potential there does not
become infinite) it cannot simply vanish within the barrier;
instead it will start to decay more or less rapidly towards zero
(Fig. T5). If the decay of the function is not too rapid the
porertrinl barrier
FIG. TS. Tunnelling of a particle from left to right through a barrier.
amplitude might not have reached zero at the outer edge of
the barrier. At this point the function must butt smoothly on
to the function characteristic of the free particle beyond the
barrier, and from then on the wave is essentially undiminished.
Since the wavefunction of the particle does not vanish in the
region outside the barrier there is a nonvanishing probability
that the particle will be found in this region, and so it has
tunnelled out of the original well.
The wavefunction inside the barrier itself falls more quickly
as the height of the barrier is raised (relative to the energy of
the incident particle), and so the amplitude of the function on
the outside of the barrier decays correspondingly quickly. This
implies that the tunnelling probability diminishes rapidly as the
barrier height is increased. The amplitude also decays more
rapidly as the mass of the particle is increased: therefore the
tunnelling diminishes as the mass of the tunneller increases. The
tunnelling
241
shape of the potential is also important because sharply changing
potentials reflect the particle more effectively than slowly
varying potentials (an analogous situation occurs in the propa
gation of light: light is reflected most strongly from regions
where the refractive index changes abruptly). Therefore the
most favourable tunnelling situation is that of a light particle
confronted with a slowly varying potential barrier. Electrons
tunnel very effectively; protons tunnel much less well, but still
appreciably; deuterons tunnel little.
Some concern is often expressed about the apparently
nonsensical occurrence of a negative kinetic energy in the classi
cally forbidden regions, where the potential energy apparently
exceeds the total energy. The occurrence of a negative kinetic
energy is indeed the reason for the exclusion of particles from
these regions in classical mechanics, but it is no problem in
quantum mechanics because the "kinetic energy of a particle
must be interpreted as the mean value evaluated over the
entire wavefunction. The fact that the kinetic energy is locally
negative does not interfere with the fact that the measurable
kinetic energy, the mean value, is positive. Furthermore,
suppose we were to attempt to confine the particle into the
region of the barrier in order to force it to have negative kinetic
energy: with drums sounding and bugles blaring the "uncertainty
principle gallops to its aid; for the very act of confining the
particle to a particular region introduces an uncertainty into
the energy, and this uncertainty is sufficient to prevent us from
concluding that the particle has a negative kinetic energy.
Questions, 1 . What is meant by tunnelling? What are the most
favourable conditions for it? What is its quantummechanical
explanation? Why is it unnecessary to worry about negative
kinetic energies? Discuss the tunnelling probability for particles
that are fired with gradually increasing kinetic energy against a
rectangular barrier. What happens when the kinetic energy is
enough, according to classical mechanics, to take them cleanly
over the top of the barrier?
2. Consider a rectangular barrier of height V and width W; on
either side the potential is zero for all space. Set up and solve
the "Schrodinger equation for this system, and from the
continuity of the wavefunctions at the edges of the barrier
construct acceptable solutions. Show that the probability of
tunnelling depends strongly on the values of V and W, and on
the mass of the particle. Calculate the relative probabilities of
an electron, a proton, and a deuteron tunnelling through the
barrier.
Further information . Calculations on tunnelling phenomena
are described in Chapter 1 1 of Bohm (1951), §26 of Davydov
(1965), Messiah (1961), Schiff (1968), and other standard
works. Problems and worked solutions will be found in
Gol'dman and Kryvschenkov (1961). Important applications
of tunnelling are in the "photoelectric effect, and so see that
entry, in chemical reactions— see Harmony (1972) for a
review— and in electrode processes; see Albery's Electrode
kinetics (OCS 14).
u
uncertainty principle. The uncertainty principle reveals that
there exist pairs of observables to which it is not possible to
ascribe simultaneously arbitrarily precise values: as one observ
able is constrained to have a more precise value so its conjugate
partner becomes more illdefined. An experiment set up to
determine the two observables simultaneously is able to
determine one precisely only at the expense of losing infor
mation about the other (see duality); and the product of the
uncertainties in the two simultaneous measurements can never
be less than a small but nonzero value of the order of h.
The most famous example of this situation is the uncertainty
inherent in the simultaneous determination of the position of a
particle along some coordinate q and its component of linear
momentum along the same coordinate p. These two observ
ables are conjugate in the sense of the uncertainty principle,
and if we can pin down the position to within a range hq
(where 8q is actually the rootmeansquare spread of the
particle's location about some point) then the uncertainty
principle demands that the value of p must be uncertain to
the extent op (again this is a rootmeansquare spread) such
that the product of uncertainties 5q6p does not have a value
of less than jm As the position is ascertained more sharply
(and 8q decreased) the spread of p must increase in order
to ensure that the uncertainty product &q$p does not fall
below s h. Conversely, if we were prepared to forgo all infor
mation about the particle's momentum so that op could be
allowed to become indefinitely large, then Sq could be made
indefinitely small, and the position determined with arbitrary
precision. Unfortunately the implication is that, although we
now know the position of the particle with arbitrary precision,
it is not possible to predict where the particle will be at any
instant later, as we can know nothing of the particle's
momentum at the moment of determining the position. Thus
the uncertainty principle eliminates the concept of a trajectory,
a concept central to classical mechanics.
Alternatively we could measure the momentum p with
arbitrary precision, but in order to preserve the uncertainty
product we should be forced to forgo all information about
the position q: this approach also eliminates the concept of
trajectory.
It should be noticed that the uncertainty principle for
position and momentum refers to components along the same
axis and puts no restriction on the simultaneous values of these
observables along different axes. Consequently the position
along x may be measured simultaneously with the momentum
p along y, and there is no inherent limitation on the precision
of the determinations. The mathematical expression of the
uncertainty principle (see Box 28) enables us to decide which
observables are conjugate; but a rough guide is that conjugate
variables consist of the coordinate and the momentum
corresponding to that coordinate.
Discussions of the uncertainty principle are often put in the
form of presenting a duffer (and long live all such duffers) who
attempts to do an experiment which will deny the predictions
of the principle; he retires, of course, bruised from the ring.
Heisenberg, whose principle this is, presented such a duffer In
order to show that all such gedanken experiments (thought
experiments) must fail. His jester used a microscope to measure
the position of the particle and, in order to do so with increasing
precision, selected one operating with ever shortening wave
242
u ncerta i nty pr i n ci pie
243
BOX 28: Uncertainty principle
Let A and B be the "operators corresponding to the
observables A and B, and let 64 and 5fi be the r.m.s.
deviations from the mean:
54=[C4 a )W> 2 ) M , &B=l<B*)<B) 2 ] K .
According to the uncertainty principle, these must satisfy
Sa5b>Ikia,b))\,
where [A. B\ = AB  BA, the commutator.
Typical uncertainty products include the following:
See text for significance of T.
length of light. But the shorter the wavelength the more
momentum each photon carries (de Broglie relation), and
since at least one photon must be scattered into the micro
scope aperture in order for the position to be determined, it is
clear that the very act of observation imparts a momentum to
the particle. An analysis of the experiment, taking into account
aperturediffraction effects and momentum transfers on light
scattering, concludes that the uncertainty product &jr6p is
indeed not less than jh (h enters through the de Broglie
relation). In a classical world the jester would laugh last,
because h would be zero and there would be no intrinsic limi
tation on the precision.
Thought experiments of this nature illustrate at an obser
vational level what the uncertainty principle reveals about the
nature of matter at a much deeper level. Momentum and
position are linked by the interpretation of the "wavef unction.
A system in a state of welldefined linear "momentum is
described by a plane wave of welldefined wavelength; but this
wave, which for a momentum kb can be written exp \kx,
corresponds to a probability distribution proportional to
I exp \kx  2 , and this is independent of x. Therefore a state
of welldefined momentum describes a particle with a com
pletely undefined position. Conversely, in order to describe
a localized particle, a "wave packet must be formed with an
amplitude large at one point and small elsewhere. This can be
achieved by "superimposing a large number of waves of
different lengths, and therefore of different momenta. Conse
quently, the sharper the wave packet we try to form (in order
to get a more localized particle) the wider the range of
momenta of the particle.
The other pairs of conjugate observables can be found by
testing whether the "commutator of their corresponding
"operators disappears: if it does not, the observables cannot be
determined simultaneously with arbitrary precision; if it
vanishes there is no restriction. Some important pairs of
conjugate observables are listed in Box 28.
The energy time uncertainty relation differs from the rest
in a subtle way: there is no operator for time in quantum
mechanics (it is a parameter, not an observable), and so the
commutation rule cannot be applied. The relation should be
viewed as a consequence of the lack of commutation of the
position and momentum operators, or equivalently, as a
consequence of the Schrridinger equation. The energy time
relation depends upon the existence of an evolution of the
system with a characteristic time t; when such a process is
present the energy levels of the system are indeterminate by
an amount 6f such that the product t8E does not fall below
^h. For stationary states, where r is infinite, the energy may
be defined with arbitrary precision; but where a state has a
finite lifetime its energy is correspondingly imprecise.
A final word may be said on cyclic systems: the uncertainty
relations for angle and angular momentum must be treated
with care because an uncertainty of 2ir in angle is equivalent
to complete uncertainty: special forms of the uncertainty
principle are used in these cases.
Questions. 1. State the uncertainty principle. Discuss the
principle as applied to the determination of the position x and
the momenta p x and p , May the position coordinates (jk, y, z)
of a particle be specified simultaneously with arbitrary
precision? May the kinetic energy and the momentum of a
particle be specified simultaneously? Why does an experiment
to determine the position of a particle interfere with the
momentum of a particle? How does the wave nature of matter
illuminate the connexion between the position and the
momentum of a particle? Why does the wave picture allow
x andp to be determined with arbitrary precision? Why is
244
united atom
the concept of trajectory alien to quantum mechanics? Why is
the energytime uncertainty relation peculiar?
2. The position q of a particle is determined to within a range
01 mm, 1 ilTt, 1 nm, 1 pm; what is the corresponding simul
taneous uncertainty in the momentum p? If the particle is
an electron, to what kinetic energy does the uncertainty in
momentum correspond? Three states in an atom decay with
time constants Ots , 1 jus, 1<T t2 s; what is the uncertainty in
the energy of the atom in each excited state? The natural
width of spectral lines is determined by the lifetime of states,
as suggested by the last part of this question. See "electron
spin resonance.
3. Use the mathematical expression of the uncertainty
principle as set out in Box 28, to investigate the limitation on
the simultaneous determination of the following pairs of
observables: x aitdp ;x and p ;p x andp ; fi^ and x ; z and S! ;
kinetic energy and Coulomb potential energy; total energy and
x; total energy and dipole moment.
Further information. See MQM Chapter 4 for a deduction and
application of the principle. For a discussion of the uncertainty
principle see Heisenberg (1930) and Ingram's Radiation and
quantum physics (OPS 3), and the standard quantummechanics
texts such as Dirac (1958), Messiah (1961), Schiff (1968),
Davydov (1965), and Landau and Lifshitz (1958a). For an
interesting account laced with speculation on mind and magic
see Bohm (1951). For an account of one aspect of the energy
time relation see Salem and Wigner (1972), and for an account
of the uncertainty principle for cyclic systems see Carruthers
and Nieto (1968).
united atom. The unitedatom method, which is used to
describe the structure of molecules, is one of a variety that
employs a correlation diagram. In general, one has a set of
orbitals or states of a system when it is in one form and a
set of orbitals or states for the system when it is in another
form, and one is interested in which states of one form turn
into which states of the other, A correlation diagram consists
of two arrays of levels joined by lines which denote the way
that a state changes into another state when the system
changes from one form to another. The most important rule
for constructing such a diagram is that lines representing
states with the same symmetry cannot cross (see "noncrossing
rule). We illustrate the technique with the idea of the united
atom; this is a lineal ancestor of the Walsh diagrams, which
show how the molecular orbitals of molecules change when
bonds are bent, and both are parents of the WoodwardHoffman
rules, which show how molecular orbitals and states change
during concerted molecular rearrangements.
The unitedatom correlation diagram takes as one set of
states the "orbitals of two separated atoms (we let the atoms
be the same and call them A), and considers how these orbitals
change as the atoms are pressed together until ultimately they
fuse into an atom of twice the atomic number, the united
atom. The energy levels of the united atom are known, so are
those of the separated atoms; what is unknown is the structure
of the intermediate object, the diatomic molecule AA.
Let us take as the simplest illustration of the method the
atoms of hydrogen; the united atom will therefore be the
helium atom. Whether or not the fusion can actually be made
to occur in practice is important for the future of mankind
but immaterial for the present discussion. Concentrate on the
1sorbitals of the separated atoms, and envisage what happens
as we squeeze the atoms together. If the atomic orbitals are
squeezed together in phase (so that a positivegoing amplitude
*of one overlaps a positivegoing amplitude of the other) the
united atom
molecule
separate atoms
FfG. U1. The united atom (He) formed from H + H, and the inter
mediate Hj.
united atom
245
process of uniting the atoms ultimately generates the Isorbital
of the helium atom, (See Fig. U1 .) The out of phase super
position of the two atoms always possesses a 'node halfway
between the two merging nuclei: when the nuclei are united,
halfway between them means through the united nucleus;
therefore the merging of the orbitals has generated a
2porbital of helium (Fig. U1). This is known to lie above the
1sorbital, and therefore we may conclude that at an inter
mediate stage the hydrogen molecule possesses the orbitals
denoted a and 0*. These are just the bonding and an ti bonding
orbitals of 'molecular orbital theory, and to them we may
apply the aufhau process in the normal way, and so arrive at
the structure of the molecule.
This process may be applied to a more complicated pair of
atoms in order to arrive at some assessment of the structure of
the intermediate molecule. There are some difficulties, and
these are mainly connected with the role of spin orbit coupling
in heavy atoms, for this serves to muddle some of the
correlations.
Questions. What is meant by a "correlation diagram'? In what
sense is the unitedatom procedure an example of the use of a
correlation diagram? To what use may the unitedatom
procedure be put? In what ways does the correlation diagram
drawn in the figure differ from the actual dependence of the
energy of the orbitals on the internuclear separation? Do
these differences matter? What rule has to be observed in the
formation of a diagram? Construct the full correlation diagram
for the formation of a united atom from two atoms possessing
1s, 2s, 2p, 3s, and 3porbitals, and use it to discuss the
electronic structure of the homonuclear diatomics that may be
formed from the firstrow atoms. (Check your answer against
Fig, M8, which shows the intermediate situation corresponding
to the structure of 2 .)
Further information. A nice discussion will be found in §4.7 of
Coulson (1961), and an original paper on the subject is that of
Mul liken (1932). See also Chapter VI of Herzberg (1950) for a
thorough discussion of the way the united atom is used to
discuss the structure and spectra of diatomic molecules. A rule
of special importance is the WignerWitmer spincorrelation
rule, which tells how to determine which atomic states are
formed when a diatomic breaks up. This is discussed in
Herzberg (loc. cit.), §5.2 of Wayne (1970), and Chapter 3 of
Gaydon (1968). The Walsh rules for the discussion of molecular
structure are given in a classic series of papers by Walsh (1953).
What these diagrams are diagrams of has been the cause of
much perplexity: for a readable analysis consult Coulson and
Deb (1971 ). The WoodwardHoffman rules are described by
Gill (1970), Woodward and Hoffman (1970), Gill and Willis
(1969), and LonguetHtggins and Abrahamson (1965). See
also Woodward and Hoffman (1969, 1970) for a review with
many applications and also Alder, Baker, and Brown (1971)
for a helpful description.
V
valence bond. The valencebond theory was the first
quantummechanical theory of the chemical "bond, and
drew heavily on the chemist's concept of a bond as an
object depending for its strength on the presence of two
paired electrons. The theory picks out of a molecule the
electrons that are paired (the perfectpairing approximation)
and supposes that these dominate in the formation of the
bond; when several perfectpaired structures have similar
energies the molecule is allowed to "resonate among them,
and the energy of the whole is thereby lowered.
As in most things, the simplest object can elucidate the
method most effectively. Consider H 2/ that most public of
molecules. At great separations the "wavefunction for the
species is the wavef unction for the two separated atoms
V> (1) ^ b (2), which for brevity we shall denote a\b 2  When
the atoms are as close as they are in the molecule the wave
function might not differ very greatly, the only difference
being that we cannot stop one electron slipping off its
nucleus and visiting the other. In other words we must permit
the electrons to exchange their roles. In terms of the wave
function we must \eta l b 2 be contaminated by a 2 b x , in which
electron 2 occupies the orbital hitherto occupied by electron
1 , and vice versa. In fact, from the symmetry of the system,
the contamination must be allowed to proceed so far that the
wavef unction is a 50:50 mixture of both arrangements
{a\bi +a 2 bi). At this point we sit back, the physics having
been done, and do the mathematics. This means that we
attempt to calculate the energy of the molecule from the
wavefunction we have set up.
There are some tricky integrals over the coordinates of the
electrons that need to be done in order to evaluate the
potential energy of the molecule, and they are illustrated in
Fig. VI , One integral can be interpreted as the contribution
j'=/d^, a a)V/4Tr£g
FIG. VI. Contributions to the v.b. energy of H 2 .
246
valence bond
247
enerWau
FIG. V2. The dependence on (he nuclear separation of the mo I ecu I ar
integrals contributing to the VB energy of H^ The total energy is also
shown.
to the total energy from the attractive interaction / between
an electron on one nucleus and the other nucleus; another is
the repulsive interaction / between the electron clouds on the
two nuclei. An analysis of the electron distribution for the
wavef unction a^ +a 2 6, shows that there is a significant
accumulation of electron density in the internuclear region
this extra accumulation of density is represented by the oval
shapes in Fig. VI— and this extra density contributes extra
terms to the total energy. One contribution is the repulsive
interaction k between the two electrons confined to the oval
BOX 29: Molecular integrals and experimental data for H 2
S = /dT (3(1)60)
/ = SdTidr 3 «(1 ) 2 (e 7 /4morn)b{2\ 2
/^fdT ia (\) 2 (e 2 /4m^ ib ) =/dri6(2) 2 (e 2 /47re r 2a )
k = fdr, dr 2 a( 1 )b{ 1 )ie 2 /ATre r 11 )a<2>o(2)
*' = /dTi3(1)6(1 )(* a /4nEoT ,„) = fdT 2 a(2)b(2) \fi 2 fA,iK^ u ).
Coulomb integral J = {e 2 f4m R) +/" — 2/'.
Exchange integral K = ie 2 S 2 /4ire R) + k 2Sk'.
Energy E ± = 2E U (H) + { jff.} (£, < SJ.
Fi,(H) is the energy of an isolated hydrogen atom.
Bond length inH 9 : 74 16 pm (07416 A).
Dissociation energy (£>g): 4476 eV, 36 116 cm" 1 .
Rotational constant: 60809 cm" 1 .
Vibrational frequency: 43952 cm" 1 .
regions, and another is the attractive energy k' between these
electron rich regions and the nuclei {see Box 29 and Fig. V2).
Just looking at the numerical values of the integrals shows that
the most important contribution is the last: in this model the
reduction in energy of the molecule below the energy of the
separated atoms is in large measure due to the lowering of the
potential energy of the electrons by virtue of their accumulation
in the internuclear region, where they are able to interact attract
ively with both nuclei. (We note and emphasize that an accurate
account of the source of bonding energy must also take into
account the changes in kinetic energy of the electrons on bond
formation and the distortion of the atomic orbitals in the
vicinity of the nuclei: see °bond and "molecular orbitals.)
The numerical value of the bonding energy for the hydrogen
molecule calculated in the manner described is 314 eV, in
moderate agreement with the experimental value 472 eV. (In
fact, the agreement is bad, but when it was first obtained the
number supported the view that the model was correct; it has
since been made much better without altering the essential
features of the description.)
248
valence bond
This approach to hydrogen is developed to account for the
structure of more complicated molecules, but the procedures
rapidly become more complex. In each case the bonding is
ascribed to the interaction of pairs of electrons, and so pairs
of electrons are selected and the energy of their interaction is
calculated. One of the important features of the valencebond
theory now appears: because there are many electrons in an
atom, and therefore many pairs, we have to take into account
the possibility that electrons that form different paired bonds
still interact electrostatically. In this connexion we may think
of "benzene and one of its Kekule structures. In the structure
the spins are paired when TTbonds are formed, and so the con
tribution of each bond to the energy of the structure arises in the
same way as we discussed for the hydrogen molecule. But the
electrons in one bond interact with the electrons in the others.
This gives rise to two effects. First, the energy of the Kekule
structure Is modified; and second, there is a tendency for the
electrons to redistribute themselves around the ring (see
Fig. B8 on p. 19 or Fig. R7 on p. 201 ). One redistribution
corresponds to the other Kekule structure, and so the effect
of the interaction is to induce "resonance between the two
Kekule structures. This alters the energy. The true distribution
of electrons in the ring cannot be described simply by the
resonance of two Kekule structures: a better description and
a lower energy are obtained if other canonical structures are
allowed to take part in the resonance (see 'benzene).
The principles described in the preceding paragraphs are
the bases of the general vafencebond method. One selects the
basic perfectpaired structures according to a set of rules (and
so they are called canonical; canon = rule), calculates the
energy of each, and then determines the energy of a super
position admitting resonance among them.
The valencebond theory has much room for improvement.
Seeing how H 2 is improved is a guide to seeing how other
molecules are improved. The flaw in the simple picture of H 2
lies in the method's implicit insistence that, if an electron
occupies an orbital on one atom, then the other electron must
be on the other atom. In practice we know that there is a
significant possibility that both electrons will be found on the
same atom, and so the wavefunction ought to be improved by
permitting an admixture of ionic contributions a\a 2 , corre
sponding to H~H t ,and/) 1 62, corresponding to H + H~. According
to the "variation principle we know that an improvement of
the wavefunction leads to a lower energy, and this is found
when an ionic component is allowed to contaminate the
covalent wavefunction. The name given to this mixing in
valencebond theory is ioniccovalent resonance because the
molecule resonates between the forms and the energy relaxes
in the normal way.
This discussion has led us to the point where we are able to
mention the disadvantages of the valencebond theory, disad
vantages that have offset the advantage of chemical plausibility
at the root of the theory. One disadvantage is that the number
of canonical structures which ought to be included increases
dramatically with the number of atoms in the molecule. For
example, there are 5 structures for benzene, but more than
100 000 for coronene. The structure must be allowed to
resonate around all these canonical forms, and so it can be
appreciated that the determination of the energy and structure
of a moderately large molecule is a task of enormous magni
tude. Another difficulty is the importance of the ionic
structures which must be added to the canonical structures. The
importance of these increases dramatically as the number of
atoms increases. The modern tendency, however, is to recon
sider these disadvantages, and present indications suggest that
valencebond theory is about to be given a second chance.
Questions. 1 . What is the inspiration for the valencebond
approach? What is the perfectpairing approximation? Why
is it only an approximation? What happens when there are
several perfectpaired structures of similar energy? What is
the consequence for the energy in that case? Give an
example of this situation in the case of H 2 and benzene.
Discuss the VB description of molecular hydrogen. What
is the nature of the reduction in the energy of the molecule
relative to the energy of the separated atoms? What is the
role of electron exchange in Hj? Why, then, are electron
pairs so important for the formation of the chemical bond?
2. The VB wavefunction for H 2 is (a 162 + 82 &i ) > n tr e
notation used in the text. Deduce an expression for the
energy of the molecule in terms of this wavefunction and
the correct "hamiltonian, and confirm that the integrals
that arise are those shown in Fig. VI and Box 29. Show that
for the alternative combination {aib 2 —g 2 bi) the energy
valence bond
249
is determined by the same integrals but with some different
signs. Assess the sign of the integrals and thence show that the
lowerenergy wavefunction is the former. What, according to
the 'Pauli principle, can be said about the spins in the bond?
If the spins were parallel (unpaired) what would we be forced
BOX 30: The VB secular determinant
1. Form all the canonical structures within the perfect
pairing approximation (for example, the two Kekule and
three Dewar structures of benzene). If then desperate with
complexity, select a manageable number of important
structures on the grounds of chemical intuition (for
example the two Kekule structures).
2. Superimpose each structure with itself and all other
structures and calculate the element H — ES , which
re re'
stands in the rth row and cth column of the secular
determinant, by applying the formula
H.
ES fc = fy N '[JE + AK]
2/V is the total number of electrons, / is the number of
islands formed in the superposition, and A is the number of
connected pairs of neighbours in these islands, less half the
number of pairs of neighbours on separate islands. We mean
by neighbours the orbitals between which there is an inter
action (and typically geometrical neighbours), J and K are
the Coulomb and exchange integrals.
An example. Let r correspond to one Kekule form, and
c to a Dewar form of benzene
r 1 + 1 H
A/ = 3,4=3+1j2 = 3,/ = 2
H rc~ £S rc = ^ £ + ^
3. Construct the full secular determinant (5 X 5 in the
case of benzene), find the roots E, select the lowest root as
the energy of the molecule, and find the coefficients in the
superposition that corresponds to this root. The squares of
the coefficients give the weights of the canonical structures.
to conclude? The variation of the molecular integrals with
distance is given in Fig. V2 for the case of H 2 . From the
values, deduce the energy and the bond length.
3. What is meant by a 'canonical structure'? How many
canonical structures are there for benzene? One way of
deducing the number of canonical structures is by means of
the Burner diagram, where all the contributing orbitals are drawn
on a circle and then pairs are joined until there are no unpaired
points. The number of structures that can be drawn in this way
without any lines crossing is the number of canonical structures
for the problem. Investigate this device for benzene and
naphthalene. Can you deduce a general formula for the number
of canonical structures for aromatic molecules? The way that
the energy of each structure and the energy of interaction
between structures is calculated is set out in Box 30. Each
canonical structure is superimposed on each other (and itself)
and the energy is related to the number of 'islands' formed by
the superimposed tines. Use the formula in the Box to deduce
the energy and interaction energy of the two Kekule structures
of benzene, set up the "secular determinant in order to
determine the energy and structure of the best (lowestenergy)
superposition, and deduce that it consists of 50 per cent of each
structure with an energy J + 24/C, and therefore that the
"resonance energy is 0'9 K. Now include the three Dewar
structures, and express the state of the molecule as
C K^ K J + ^' +C D^ D) + ^D2 + ^D3l' inan0bUiOUS
notation (I hope); set up the appropriate secular determinant,
and deduce the extra stabilization energy that arises from
admitting the Dewar structures. For what proportion of the
structure do they account?
Further information. See MQM Chapter 9. A nice account of
simple VB theory, and an extensive comparison with molecular
orbital theory is given in Chapter V of Coulson (1961 ) and by
Murrell, Kettle, and Tedder (1965). See Eyring, Walter, and
Kimball (1944) for details of the method. Pauling's classic
book (1960) is almost exclusively an account of the VB
description of molecules and in its earlier editions is an un
paralleled example of the power of quantum chemical reason
ing within the format of the theory. The molecularorbital
and valencebond theories are discussed in comparison under
"molecular orbital versus valence bond.
250
valence state
valence state. Let us centre our attention on carbon, and in
particular on the carbon atom in methane. A chemical descrip
tion of the structure of methane might regard it as an
sp 3 "hybridized carbon with each of its four tetrahedral lobes
overlapping one of the four surrounding hydrogen atoms.
Therefore the bonds are formed from four hydrogen atoms
overlapping the four tetrahedral orbitals of a carbon atom in
its valence state 1s 2 2s2p v 2p i/ 2p The valence state is the
X y £
state of the atom responsible for its bonding to its neighbours.
From this definition it is a trivial consequence that the valence
state of the hydrogen atom is simply 1s; similarly the valence
state of the oxygen atom is 1s 2 2s 2 2p^2p, 2$^ in a substance
such as water.
It is important to realize that the valence state is not in
general a spectroscopic state; that is, it cannot be detected by
lines in a spectrum representing transitions into or out of it.
The clearest way of appreciating this is to reflect on the nature
of the bonding between the carbon and the hydrogen atoms
in methane, and to recognize that although the electrons are
"paired in individual bonds there is no pairing of spins in
different bonds. Therefore the valence state is characterized by
the four valence electrons with random relative spins. But a
state with random spins is not a spectroscopic state, for in
these there is a strict coupling of the various angular momenta,
and therefore a strict distribution of relative spin orientations.
It is of interest to be able to know the energy of the valence
state, especially when the importance of "hybridization is being
assessed; this may still be done even though the state is not
spectroscopic. The valence state may be expressed as a mixture
{'superposition) of true spectroscopic states, and its energy
calculated by the corresponding average of the energies of the
contributing states. In this way it is possible to assess the energy
required to promote an electron from the ground state of
carbon to form the valence state {7 eV, 680 kJ mol"' ).
Is the valence state ever formed? Since it is not a spectros
copic state the answer is strictly no (but see Questions); but we
may envisage the valence state as emerging from the ground
state as the ligand atoms are brought up towards it. Once again
consider methane, but a 'potential methane' in which the four
hydrogen atoms are disposed tetrahedrally at infinite separation,
and a central ground state carbon atom. As the atoms approach
tetrahedrally the surface of the carbon atom begins to stir, and
for brief moments the electron density might tend to accumu
late tetrahedrally. As the atoms get even closer the fluctuations
are stronger and more longlasting; and when the atoms are at
their equilibrium bonding distance the fluctuations are massive
and essentially frozen, forming the four tetrahedral abonds.
Only at this point would it be true to say that the central atom
was in its valence state, which has been drawn out of the ground
state by the presence of the hydrogen atoms and the bond
energy that lowers the energy of the whole system.
Questions. What is a valence state? Why is it not a spectroscopic
state? How may the energy of the valence state be determined?
What role does the valence state play in chemical bonding
theory, and why is its energy important? The valence state can
be expressed as a superposition of spectroscopic states; suppose
we contrived to produce an atom in a state which was just such
a superposition of spectroscopic states, discuss the history of
the state from its moment of formation.
Further information. See MQM Chapter 9. See also Coulson's
The shape and structure of molecules (OCS 9) and §8.4 of
Coulson (1961 ), who gives the following references to
calculations on valence states, their composition, and their
energy: Voge (1936), Mulliken (1938), Pauling (1949), Moffitt
(1950), and Skinner (1953, 1955). A helpful discussion of
valence states, with examples, is given in §4.1 of McGlynn,
Vanquickenborne, Kinoshita, and Carroll (1972).
variation theory, or variation principle. The energy
calculated using an arbitrary "wavef unction cannot be less
than the true lowest energy of the system.
In quantum theory we are told to calculate the energy of a
system by evaluating the "expectation value of the "hamiltonian
of the system, and so we evaluate the quantity E=
JdT ^W^/Jdr^*^. The variation principle informs us that
if we make an arbitrary choice of the function <f> then the
analogous quantity 8= fdT<p*HigffdT<p*<p, which is called the
Rayleigh ratio, cannot be less than the true groundstate energy.
The implication of this important result is that if we make a
series of guesses about the form of the trial function, the one
that gives the lowest energy will most closely resemble the true
wavefunction of the system. If we are lucky we shall guess a
function that yields the true energy: in that case we shall have
vector model of the atom
251
found the true groundstate wavefunction. It is important to
develop a method of choosing the best function other than
relying on mere intuition, and two techniques are often
employed.
The first writes the wavefunction as a function of one or
more parameters, and then varies the parameters in search of
a minimum. Thus if the trial function were dependent on the
value of a parameter p, differentiation of £{p) with respect to
p and determination of the condition for a minimum yields the
best value of p and therefore the best function of that particular
form. Of course we might have chosen a function of a poor
form, but the function so found would be the best of that type.
As an example, we might have guessed that the ground state of
the "hydrogen atom was welt described by a function exp{— pr 2 ):
a variation treatment leads to a best value of p, but not a very
good energy. A function exp(— pr) requires another value of p,
and in this case the best energy is the exact groundstate energy,
and therefore the trial function with this value of p is the exact
groundstate wavefunction.
A different approach was introduced by Ritz: he supposed
that the trial function could be written as a sum of functions:
the functions themselves are invariable, but the amounts of
each in the mixture constitute the variable parameters. The
trial function is of aformv3=Pi ^i +p 2 2 + , . . ,and the
minimum is found by differentiating &(p t ,p 2 , , . . ) with
respect to all the parameters p f and seeking a simultaneous
minimum. This procedure is the basis of the method of deter
mining the best mixture in the method of "linear combination
of atomic orbitals. Once again the minimum energy differs
from the true energy if an insufficiently flexible trial function
has been chosen: it also differs if an approximate hamiltonian
has been used (in which case an energy below the true energy
may be found, for the hamiltonian must be correct if the
variation method is to be tried).
It would be very useful to know how far the variational
minimum energy, which is an upper bound to the true energy,
lay above the true energy. There are techniques of finding a
lower bound (beneath which the true energy cannot lie), and
in principle this gives some indication of the accuracy of a
variational calculation; but the technique is difficult and has
not been widely used.
Questions. 1. State the variation principle. Does it provide an
upper or a lower bound to the true energy? If one guesses a
wavefunction and calculates the energy of the system with it,
of what can one be sure? What are the two methods of
selecting the best trial function of a particular form? Why
might the energy so calculated still be considerably greater
than the true energy of the system? What is the Ritz procedure?
What should be determined in order to estimate the error in the
variation calculation?
2. Take a trial function of the form exp(— pr) and vary p to
find the ground state of the hydrogen atom. (The form of
the "hamiltonian will be found in Box 7 on p. 90 and the
radial part of the 'laplacian in Box 11 on p. 124). Do not
forget to maintain the normalization of the function; in
other words, minimize the ratio fdTip*H$fdT<p*tp with
respect top. Repeat the exercise with a trial function of
the form exp(— pr 2 ). Now try a function of the form
piexp(— pit 2 ) +p 2 exp(— piT 2 ) and attempt to achieve a
lower minimum. Sketch the form of the three best trial
functions.
3. The Ritz procedure takes a trial function of the form
?p.i/<. and varies the parameters p.. Show that the condition
fit I
for a minimum energy is attained when the determinant
I W  SS.. vanishes. H.. are the integrals fdT^/^H^f and 5 ( ..
the integrals fdrify. *^. The minimum energy is the smallest
root of the determinant, and the determinant itself is known
as the secular determinant. Apply the Ritz variation principle
to the demonstration that the Isorbitals in molecular hydrogen
contribute equally to the bonding molecular orbital.
Further information. For a simple proof of the variation
principle and a derivation of the minimum conditions, see
MQM Chapter 7, and §3.6 of Coulson (1961). For further
discussion see §7c of Eyring, Walter, and Kimball (1944),
Kauzmann (1957), Pilar (1968), and Wilcox (1966). For a
discussion of the determination of lower bounds see Lowdin
(1966) and references therein.
vector model of the atom. The vector model is a represen
tation in terms of vectors of the coupling of the angular
momenta of the electrons of the atom. The basis of the method
vector model of the atom
is the representation of a state of "angular momentum of
magnitude [/, (/, + 1(j *h by a vector j, of length [/, (/, + 1 )] *
with an appropriate orientation. If the component of
angular momentum on some arbitrary zaxis is well defined
and has the value mh the orientation of the vector is drawn so
that its component on this axis is of length m. Since such an
angular momentum "processes about the 2axis, the vector ts
drawn so that it lies on a cone at some arbitrary but indeter
minate azimuth (Fig. V3).
a
lenqrh /JCJ4D'
classical trajectory
FIG. V3. The basic vector model of angular momentum.
If a second source / 2 is present, the total angular momentum
of the system may be constructed as the resultant of the two
vectors representing the two momenta: since the length of the
resultant vector must be [/(/ + 1 )] w , with the value of/ selected
from the set/, +/ 2 ,/, + /i  1 , . . . I/, j 2 \ (see angular
momentum), only a few orientations of the three vectors
ii.h. and their resultant j are permitted. In accord with the
algebraic theory of angular momentum the vectors j i and j 2
precess around their resultant, and the latter precesses around
some arbitrary zaxis. This situation is represented by a vector
diagram of the type shown in Figs. V4 and V5. The process of
coupling the momenta together may continue until all the
individual spin and orbital contributions have been combined
into the one resultant representing the total angular momentum
of the whole atom. Fortunately this formidable exercise can be
simplified by a number of approximations. The first simplifi
cation is to note that the core of the atom (the electrons in
other than the valence shells) has zero angular momentum
because all its spins are paired and the shell is complete. The
FIG. V4. Coupling of j i and jj in the vector model.
FIG. V5. Coupling of j t =2 and/ 2  1 to give/ = 3, 2, 1. A vector
construction is shown in black, and a simple rule of thumb, which
uses lines of length Jt.hJ. is shown in colour.
next approximation supposes that there are two extreme cases
of coupling.
1 . The first, the RussellSaunders coupling case or LScoupl/ng
case, assumes that the °spinorbit coupling is so small that it is
effective only after all the orbital momenta have been summed
into some resultant L, and all the spins summed into another
resultant S. The electronic orbital motions are dominated by
{
vector model of the atom
253
the electronelectron electrostatic interactions, and this is the
source of their coupling energy. For two electrons the coupling
of I j and [ 2 would be represented by a diagram of the type in
Fig, V6 a. The two spins also couple to give a definite resultant
S; the coupling energy for this interaction arises from the
spin dependent 'exchange energy, and so it too is an electro
static phenomenon. At this stage the two resultant momenta
L and S couple together to form a resultant J, the total angular
elecrrastaric coupling
S.
•
Spinorbif coupling
clecrrosforic coupling
FIG. VG. (a) RussellSaunders or LScoupling. (b) //coupling.
momentum, and the strength of this interaction depends on
the strength of the spinorbit coupling.
2. When the spinorbit coupling is stronger than the electro
static interaction the Russell Saunders scheme breaks down
because the spin and orbital angular momenta of individual
electrons attempt to organize themselves into satisfactory
mutual orientations. The jf coupling scheme describes the
extreme situation of this kind. In it each electron's spin is
allowed to couple with its orbital momentum; thus l t and
Sj couple to form Ji . The two components "process strongly
around their resultant in the manner characteristic of a
strongly interacting pair of momenta. This jj is now coupled
to another j 2 , and the total angular momentum J constructed:
the latter coupling is relatively weak, for it depends on the
electrostatic interactions of the electronic orbitals. We see
that, although the total angular momentum obtained in this
way might be the same as the total in the Russell Saunders
scheme, the states of the atoms are different, and their
energies also differ.
Neither scheme is an exact representation of the true state
of affairs because there is always some competition between
the different types of interaction, and indeed it is quite
possible for some electrons in the same atom to be coupled by
one scheme and the remainder by the other. Nevertheless
for light atoms, which have small spinorbit coupling
constants, the RussellSaunders scheme is often a good
description of the valence electrons. Heavy atoms, which have
large spin orbit coupling constants, are often predominantly
//coupled. It follows that the wavef unctions corresponding to
the Russell Saunders scheme are a good starting point for
more elaborate calculations on light atoms.
The angular momenta that one is led to by the vector
model are the bases of the labelling of the state of atoms by
"term symbols.
Questions. What is the vector model of the atom? What are
its basic features? What is the length of the vector that would
represent the orbital angular momentum of an eiectron in an
sorbital, a porbital, and a dorbilal? What is the length of the
vector representing the "spin of an electron? Construct vector
diagrams for the coupling of the spin and orbital angular
momenta of an electron in a porbital. What is the energy of
254
vibrational spectroscopy: a synopsis
interaction? What approximations may be introduced to
simplify the discussion of the coupling of angular momenta in
atoms? By a vector construction show that the angular
momentum of a complete Kshell (Is 2 Ms zero. What is the
source of the coupling energy between the orbital momenta
of electrons? What is the source of the spin coupling energy?
When is it appropriate to use Russell Saunders coupling? When
should //coupling be used instead? Are there any alternatives?
Is it possible to use the RussellSaunders term symbols even
when the//coupling predominates?
Further information. See MQM Chapters 6 and 8 for a further
discussion of the vector model of the atom and the two
coupling schemes. Good accounts are given in §1 1 .2 of
Herzberg (1944), Chapter V of White (1934), §3.4 of King
(1964), and Chapter 12 of Kuhn (1962). Candler's book
(1964) concentrates on the vectormodel description of atoms.
"Hie molecular situation is outlined in the section on the Hund
coupling schemes.
vibrational spectroscopy: a synopsis. As a first approxi
mation the vibrations of molecules are assumed to be simple
•harmonic. The frequency depends on both the "forceconstant
and the mass of the vibrating object according to o>= (it/m) H
in radians per second. Molecular forceconstants and masses are
such that vibrational frequencies fall in the infrared region of
the spectrum, and so obtaining a vibration spectrum is an infrared
spectroscopic technique. Vibrational structure also appears in
an "electronic spectrum, for during an electronic transition
vibrations may be excited: the spacing of the lines of this
vibrational structure is of the order of an infrared frequency.
As a rough guide, weak bonds between heavy atoms vibrate in
the region of several hundred cm" 1 {the I I bond in \ 2 vibrates
at 214 cm" 1 , 64 X 10 11 Hz, or 467 X 10 4 nm, in units of
wave number, circular frequency, and wavelength respectively),
and stiff bonds between light atoms vibrate in the region of
several thousand cm" 1 (the HH bond in hydrogen vibrates at
4395 cm" 1 , or 13 X 10 w Hz, or 2280 nm). Bond stretches
tend to be at higher frequency than bond bends. See Table 10
for the vibrational frequencies of some diatomic molecules,
and Table 24 for the typical frequencies of groups in molecules.
A vibration is activei.e. observable in the infrared if
during it the dipole moment of the molecule changes. This
implies that diatomic molecules absorb in the infrared only
if they are polar. In more complicated molecules it is
necessary to scrutinize the form of the 'normal mode in
order to see whether the vibrations of several atoms
jointly lead to an oscillating dipole. This may often be done
by inspection, but more positively one should take into
account the symmetry of the system by using "group theory.
If the vibration is active the selection rule for the transition is
that the vibrational quantum number for that transition may
change by ± 1 . At normal temperatures the Boltzmann
distribution ensures that essentially all molecules are in their
ground vibrational state: this implies that the spectrum should
consist of a single line for each vibrational mode of the
molecule, and correspond to the excitation of the mode of
vibration from its ground state to its first excited state. Such
a line is the fundamental and is denoted (I* — 0).
The vibration of a molecule is not strictly harmonic because
the potential in which the atoms move is not strictly parabolic:
the deviations are greater at large displacements from equilib
rium. This 'anharmonicity has several consequences. First, the
selection rule for the transition fails to a degree that depends
on the amount of anharmonicity. Instead of a single line for
each mode one sees the fundamental <1<— 0), or the first
harmonic, accompanied by weaker overtones, or second,
third, . . . harmonics corresponding to the 'progression of
transitions (2*— 0), (3* — 0), . . .. These should appear at the
frequencies 2w, 3cj, ... but not exactly because of the
anharmonicity. It is possible for two modes of a molecule to
be excited simultaneously if they are not entirely independent
(that is, if there is present anharmonicity which is able to mix
together the two modes) : the absorption that is responsible for
this appears as a combination band. When the energy of a
combination level lies close to the energy of an unex cited
mode, which may be unexcited because it is inactive in the
infrared, there may occur a 'resonance interaction between
them by virtue of the anharmonicity present. This Fermi
resonance causes the lines to shift, and the active bands donate
intensity to the inactive, which therefore appear in the spec
trum (this is intensity borrowing, brought about by the inactive
vibronic transition
255
mode acquiring some of the properties of the active modes :
see "superposition principle).
When a vibrational transition occurs it may be accompanied
by a "rotational transition of the molecule. This gives rise to a
structure in the spectrum which is observable when the sample
is gaseous; in a liquid collisions with the solvent blur the
structure by reducing the lifetime of the rotational states. In
an electric dipole transition the "rotational "quantum number
J may change by or ± 1 ; consequently there are lines at the
position of the pure vibrational transition (A/= 0) which
constitute the Q'branch of the spectrum; a series of lines at
lower frequency (/\J = 1 ), the Pbranch, and a series at higher
frequency (A J = + 1 ), the R branch. A series of lines rather
than a single line is observed because the Boltzmann distri
bution permits a number of rotationai levels to be occupied
in the initial state, and each one of these gives rise to a line
in the branch. The branches may pass through a head if the
rotational constant of the upper vibrational level is signifi
cantly different from the rotational constant in the lower
level (see "branch); this is especially important when the
vibrational transition is part of an electronic transition for then
the rotational constants may be very different.
The other features that affect a vibrational spectrum in
clude "inversion doubling (for example, in NH 3 ) and ttype
"doubling in linear tri atomic molecules. See "Coriolis inter
action.
The main pieces of chemical information that one may
obtain from a study of vibrational spectra include the elemen
tary but important one of the identification of a species by
using its vibrational spectrum as a fingerprint. The major quan
titative information that may be obtained is the rigidity of
bonds under the stresses of stretching and bending: the
forceconstant is an important feature of a chemical bond. The
anharmonicities show how far the true potential differs from
an ideal parabola. The rotational structure on vibrational
transitions enables the molecular geometry to be determined
in different vibrational states (bondangle and bondlength
dependence on vibrational state), and the vibrational and
rotational structure of electronic transitions enables the same
kind of information to be obtained about electronically
excited states. This information enables one to build up a
full picture of the potential energy curves of molecules in
different electronic states.
We have concentrated on electric dipole absorption spectra:
vibrational transitions may also be observed in "Raman
spectroscopy.
Further information. See MQM Chapter 10 for a discussion of
vibrational and rovibrational spectra in more quantitative
terms, and with the use of group theory. An introduction to
the vibration of molecules may be found in Barrow (1962),
Whiffen (1972), and King (1964). The characteristic frequencies
of many bonds are listed in Bellamy (1958, 1968) who also
describes infrared spectroscopy as an analytical tool. More
advanced discussions are given by Gans (1971), Herzberg
(1945), Wilson, Decius, and Cross (1955), and Allen and Cross
(1963). See also Gaydon (1968).
vibronic transition. The word 'vibronic' is an amalgam of
w'6ration and electrowc, and implies that the transition involves
both modes of excitation simultaneously, A vibronic state is
the name applied to a state of the molecule when it is improper
to view the electronic and the vibrational states as independent.
In order to elucidate this description consider an octahedral
complex and an electronic transition of a delectron. In the
"crystalfield or "ligandfiefd theories of transition metal
complexes the dorbitals are split into two groups sep
arated by a small energy difference; therefore it is tempting
to ascribe the colours of transitionmetal complexes to a
transition of an electron from one set of dorbitals to the other.
The problem that immediately confronts us is the Laporte
"selection rule, which forbids dd transitions because it forbids
"geradegerade (gg) transitions. Most rules can be evaded, and
one of the rules for looking for ways of evading rules is to
seek the approximation on which that rule might be based,
and then to repair the approximation. The Laporte selection
rule is based on the existence in the complex of a centre of
symmetry, and only if the complex is strictly octahedral is the
rule strictly valid. But the complex may vibrate, and some of
the vibrational modes destroy the centrosymmetric nature of
the molecule. Now consider the unexcited, vibrationally
quiescent molecule, and a photon approaching it. Let the
256
virial theorem
photon excite simultaneously a delectron and a vibration of
the complex. Then if the excited vibration is one that destroys
the centrosymmetry of the complex, the Laporte rule will
be slightly but sufficiently broken, because the complex no
longer possesses a centre of inversion in the initial and final
states.
The transition is allowed only in so far as it is proper to
treat the vibration and the electronic motions as coupled
together so that they jointly determine the symmetry of the
object with which the light is interacting. Therefore the
transition is vibronic and the states of the complex must be
treated as vibronic states. This view leads to another way of
looking at the nature of a vibronic transition. In this we
consider the possibility of a transition from a d orbital to a
porbital; this, being a g to u transition, is allowed. But why
should the upper level be a porbital, or at least why should it
possess some pcharacter? If the electrons follow the nuclear
vibrations the electronic distribution in the upper state must
follow the nuclear motion. To do so it must distort from the
eentrosymmetric distribution which dorbitals give rise to,
and one way of achieving the distorted distribution is to mix
in ('hybridize) some porbital character (Fig. V7). Therefore,
when such an antisymmetric mode is excited the electronic
distribution contains some pcharacter; and as dp transitions
are allowed, the transition from the quiescent ground state to
the vibronic upper state is allowed in proportion to the
amount of pcharacter the vibration introduces.
Further information. See MQM Chapter 1 for a further
discussion, and some group theory. Vibronic transitions are
described in more detail by Orgel (1960), Figgis (1966),
Ballhausen (1962), and Griffith (1964). When the rotational
states of a molecule must be considered too then a
rovibronic transition occurs, and the composite states are
rovibronic states.
virial theorem. In its simplest form the virial theorem states
that if the potential energy of a system follows a Mr law then
the mean kinetic energy Is related to the mean potential energy
by the expression < 7"> = — ~<V>. This in turn implies that
the total mean energy of the system is simply equal to ^<.V>,
or to — <T>. A more general form of the theorem states that
if the potential energy follows the tawf", then the mean
potential energy and the mean kinetic energy are related by
<T> = ^n<V>. Thus for an harmonic oscillator, in which
the potential energy is parabolic (r> = + 2), the mean potential
and kinetic energies are related by < T> — < V>, and this
equality is yet another manifestation of the peculiarly high
degree of symmetry of the 'harmonic oscillator. The theorem
applies to a bounded system, and one that is stationary in
time; but if by mean value is also implied a time average, then
it also applies to no n stationary states.
The name 'virial' (which is derived from the Latin vis, vires;
force, forces) comes from the classical mechanical form of
the rate of change of the mean value of the product r.p, where
r is a position coordinate and p is the linear momentum. This
leads to the equation < T> =  j< F.r>, the virial theorem of
Clausius, where F is the force acting. From this expression may
be deduced the general form of the equation of state of a
real gas in which forces operate between the gas particles: this
gives rise to the virial expansion and the virial coefficients of
thermodynamics. The virial theorem may also be derived
quantum mechanically and applied to a discussion of the
structure and properties of atoms and molecules. For example,
it is a test of the exactness of a calculated wavefunction that
FiG. V7. An asymmetric vi
bration hybridizes p and d i and
permits a dd transition via the dp
component.
virtual transition
257
the expectation values of the potential and kineticenergy
operators do indeed satisfy the virial theorem. It is essential to
remember that the virial theorem imposes a connexion between
the way that the kinetic and potential energies vary as wave
functions are distorted. The virial theorem may also be used as
an alternative to the variation theorem in some circumstances.
A further generalization of the virial theorem may be made
and the hypervirial theorems obtained. These are a set of
theorems based on the vanishing of the average value of the
"commutator of an operator with the harniitonian of the
system when the system is in an eigenstate of the hamiltonian.
Questions. From the wavef unctions given in Tables 1 1 and 1 5
demonstrate the validity of the virial theorem for the ground
states of the harmonic oscillator and the hydrogen atom.
Return to the calculation of the ground state of the hydrogen
atom in terms of the variation principle, and investigate
whether the viria! theorem is satisfied for the best gaussian
approximation.
Further information . For a good discussion of the classical
virial theorem, its deduction, and some applications, see
Goldstein (1950); for the deduction of the gas laws see
Lindsey (1941). The quantummechanical virial theorem is
derived by Hirschfelder [1960), who also deduces and
describes the hypervirial theorems. Application of the
theorem to molecular and atomicstructure calculations are
described by Coulson (1965), Deb (1973), and Feinberg,
Ruedenberg, and Mehler (1970).
virtual transition. When a system is 'perturbed, for
example by the application of an electric field, it is distorted;
the distorted system can be described by a superposition of
the wavef unctions of the states of the original system, and
therefore the system behaves as though it contained features
of the excited states; it has made a virtual transition to the
excited state. When a light wave scatters from a molecule the
distortion it induces can be envisaged as a series of virtual
transitions to the excited molecular states caused by a virtual
absorption of a photon. The distortion is immediately released,
and the photon flies off leaving the molecule either in its
original state (Rayleigh scattering) or in one of the excited
states populated by the initial virtual transition (Raman
scattering).
As the frequency of the incident light approaches one of
the transition frequencies of the molecule the transition
gradually loses its virtual character and becomes real: the
molecule is really excited and the photon is really absorbed.
Energy is conserved in real transitions; but as 'virtual transition'
is just a name for a way of describing a distortion and of taking
into account the effect of a perturbation, for them it is not
conserved.
Questions, What is a virtual transition? In what ways does
it differ from a real transition? When does a virtual transition
take on the character of a real transition? Is energy conserved
in a virtual transition? What virtual transitions are involved
when an electric field is applied to a "hydrogen atom?
Further information. See MQM Chapter 7 for a discussion of
perturbation theory and a concomitant discussion of virtual
transitions. Books that deal with "perturbation theory perforce
deal with virtual transitions, although the term is not always
used. See Davydov (1965), Dirac (1958), Landau and Lifshitz
(1958a), Messiah (1961), and Schiff (1968). See §16 of Heitler
(1954) and §7.5 of Hameka (1965) for a discussion of the
transformation of virtual into real transitions. See Raman
effect and Stark effect.
w
wavef unction. The wavefunction for a system is a solution of
its Schrodinger equation and is the function that contains all
the information about its dynamical properties. If the wave
function that describes the state of the system is known, all the
observable properties of the system in that state may be
deduced by performing the appropriate mathematical
"operation. The wavefunction may be a function of time, and
is then often written ^(r,t). When it is not a function of time
(or when the timedependence has been factored out) it is
often denoted i^, and is a function of all the coordinates of
all particles that make up the system. Since the wavefunction
depends on the state of the system it is often labelled with an
index or set of indices {the "quantum numbers) that distinguish
the state. Thus the wavefunction for a system containing N
particles and needing M quantum numbers is the mathematical
function ty n (r! ,r 2 . . . r w ). As examples of wave
functions we may point to the wavefunction for a free particle
travelling in the x direction with a 'momentum kh, which is
the function exp \kx, and to the wavefunction for the ground
state of the "hydrogen atom, which is the simple function
exp(— r/a ). The wavefunction has an interpretation, must obey
some restrictions, and contains information. We describe these
aspects below.
1. Interpretation of the wavefunction. We concentrate on a
system containing one particle with the coordinate x. The
Born interpretation of $W is that it is the amplitude for
the probability distribution of the position of the particle.
According to this interpretation the probability of finding
the particle in the infinitesimal range dx surrounding the
point* is proportional to i^*(x)^(x)dx. The probability
density at the point x is therefore proportional to
i/<*(x)i//(x). If we were dealing with a threedimensional
system the wavefunction t//(r) would be interpreted as follows:
i//*(r) \jj[f)dT is the probability of finding the particle in an
infinitesimal volume element tiT surrounding the point r. The
interpretation may be pictured in terms of inserting a probe
sensitive to the presence of the particle, and which samples a
volume dT in the system; as the probe is moved to different
points r the meter reading is proportional to the volume of
the probe and to the value of \jj*{r)\p{r). As an example, the
wavefunction for the "hydrogenatom ground state is a
decaying exponential function of r; therefore the meter reading
for the electron density will fall according to exp(— 2r/a )dT as
the probe of volume dT is moved out along a radius. In the case
of the other wavefunction referred to above (expifcx) the meter
would give the same reading wherever the probe is inserted
because {expifcx)*{exptfrx) = 1 and is independent of x. This
function corresponds to an even spreading of the particle
throughout the universe, whereas for the hydrogen atom the
electron is densest close to the nucleus.
2. Limitations on the wavefunction. If the wavefunction is to
be interpreted as an amplitude for the probability density for
the distribution of the particle is must be constrained in a
variety of ways.
(a) It must be finite everywhere, for otherwise there would
be an indefinite accumulation of probability density at the
points where it became infinite. (This requirement is really too
stringent: all we need to impose is the condition that the total
258
wavef unction
259
probability of the particle being within the universe is unity—
thus we require the existence of the integral fdr\j/* (r)\p(i); but
our toostringent requirement is a good guide in most cases.)
(b) The probability density must be singlevalued every
where, because it would be nonsense to say that the prob
ability density at a certain point is both 02 and 04, In most
cases {systems involving "spin are exceptions which are easily
accommodated in another way) this requirement is the same
as requiring the wavef unction itself to be singlevalued.
(c) The wavefunction must be continuous, for it would be
unreasonable to have a probability density of a particular
value at a point and a finitely different value an infinitesimal
distance away.
The imposition of these limitations on the wavefunction
is severe, for it forces it to obey certain boundary conditions,
and leads ineluctably to quantization (see "quantum) because
only a very few of the solutions of the Schrd'dinger equation
survive when the conditions are imposed.
3. Information in the wavefunction. We have already seen that
the wavefunction contains the information about the distri
bution of the particle. The mean gradient (slope, first
derivative) of the function is the 'momentum of the particle
in that state; this emerges from the quantummechanical
rules about interpreting observables by 'operators. The mean
curvature (the second derivative) is the kinetic energy of the
state. The value of any observable is determined by calculating
the "expectation value of the corresponding "operator using
the appropriate wavefunction.
4. Timedependent wavef unctions. If the wavefunction
^„ (x) corresponds to an energy ( eigenvalue) E the time
dependent form of the wavefunction ^ {x,t) is simply the
product 4> n (x)exp(if n t/fi). This is a stationary state (even
though t occurs) because the probability density [^ (x,t)\ 2
is independent of time.
5. Pure states and superpositions. If it is certain that a
system is in a state with welldefined quantum numbers then
the wavefunction is that of a pure state. As an example, a
hydrogen atom known to be in its ground state is in a pure
state, and a particle with precisely defined momentum is also
in a pure state and is described by a simple wavefunction.
When the state of a system is believed to be one of a range
of pure states, for example, if the particle has a momentum
somewhere in the range (k — K)n to {k + K)h, then the wave
function for the system is a superposition of the purestate
functions covering this range. Thus if the state is believed to
be in the range of states spanned by the functions
^ n M< ^„ W ■ ■ ■« tne true state o f the system is described
i "2
by the linear superposition 0(x) = c,0 {x)+c 2 \p [x)+ . . .,
where the coefficients determine the probability that the
system is in one of the basis states: the probability that the
system is in a state described by the wavefunction \p (x) is
proportional to c*c n , or jc p . The coefficients may be
timedependent. An example would be the excitation of a
hydrogen atom by incident radiation: initially the atom is in
the ground state described by the function $ (f), but as
irradiation continues it takes on more of the character of the
2p^ state. Therefore during irradiation its state is described by
the function *(r,f) = c 1t W^,,W + c 2p (f)0 2p (r), with
c 1s (0)  1 and c 2p (0) = 0, and the probability that at a time
f it has actually made the transition to the 2p state is
proportional to k, (f)*. The calculation of the coefficients
is a task for perturbation theory, and the example is described
further under transition probability. An example of a static
superposition wavefunction is that of a hybrid orbital, and
another is an LCAO molecular orbital.
Questions. How is a wavefunction obtained? What is its
interpretation? What is the difference between probability
and probability density? Sketch the meter reading for an
electron sensitive probe when it is dipped into a hydrogen
atom and pushed in towards the nucleus along a radius:
first let the probe be a minute volume element, roughly a
cube of volume dxdydz, and then let the probe be a spherical
shell of area 4w 2 and thickness dr (r is the radius, and so the
sheil gets smaller the closer it is pushed towards the nucleus:
see radial distribution function). What are the three
constraints on the wavefunction? Is a wavefunction of the
formexp(+ax) a likely candidate for a wavefunction for a free
particle? What about the function x/x? A particle is
confined to a ring and the function exp im<p is proposed for
its wavefunction: what limitation must be put on the values
of m? How is a timedependent wavefunction for a stationary
state formed? Suppose the energy E were replaced by the
260
wave packets
complex quantity E— iftT; what would happen to the amplitude
of the hitherto stationary state, and how could this be
interpreted? How should the coefficients of a superposed
wavefunction be interpreted? A wavefunction is written
^ + 3 % i^ ; what proportion of s and pcharacter does it
contain? Form an sp hybrid.
Further information. See MQM Chapters 1 , 2, and 3 for a
detailed discussion of the solution of the Schrodinger equation
for a variety of systems and an account of the properties and
significance of their wavefu net ions. For a discussion of the
interpretation of wavef unctions see Pauling and Wilson (1935),
Landau and Lifshitz (1958a), and Schiff (1968). For questions
about its interpretation see Bohm (1951), Jammer (1966), and
Ballantine (1970). We have discussed elsewhere the question of
the 'normalization of the wavefunction, and the question of
■orthogonality. Its interpretation as an "cigenfunction of
the "hamiltonian is important. See also the "superposition
principle, "atomic orbitals, and the "hydrogen atom.
wave packets. A particle that experiment or observation
shows to be confined to a very small region of space must
be described by a "wavefunction that is strongly peaked
within the region and virtually zero elsewhere. A wave
function corresponding to a sharply defined "momentum
has a welldefined wavelength, and so spreads over a large
region (actually the whole) of the system; the only way of
attaining localization is to take a ^superposition of the
latter functions and investigate their mutual interference.
If the superposition has been well chosen all the construc
tive interference occurs at a selected point and destructive
interference eliminates the amplitude of the wavefunction
everywhere else (Fig. W1). The square of the sharply
peaked function is another sharply peaked function, and
so the probability of finding the particle differs from zero
only in the pointlike region. Thus a wave packet describes
a localized particle (and because we have a superposition
of a vast number of energy states the momentum is
correspondingly indefinite).
The wave packet also moves, because all the component
functions are timedependent and the point of maximum
constructive interference moves. It is possible to show that
FIG. W1. Formation of a wave packet: (a) single momentum state, no
localization; (b) several states, some localization; (c) many states, good
localization; (d) infinite number of states, perfect localization,
when the wavef unctions are the solutions of a Schrodinger
equation for a specified potential the motion of the wave
packet corresponds very closely to the motion predicted
for a classical particle in the same potential. Thus we see
how the structure of quantum mechanics underlies the coarse
description provided by classical mechanics. One important
difference is that the wave packet tends to spread with time,
but this tendency is very small for massive, slow particles.
Further information. The formation, significance, ana
motion of wave packets are described in MQM Chapter 3,
and made quantitative in Appendix 3.1, A good and instruc
tive discussion will be found in Chapter 3 of Bohm (1951),
and those with tough teeth should consult Gofdberger and
Watson (1964), especially Chapter 3.
Wigner coefficient. A Wigner coefficient, or CtebschGordon
coefficient, or vectorcoupling coefficient, is the coefficient
work function
261
in the expansion of a state of coupled angular momentum in
terms of its uncoupled components.
As a specific example, consider the coupling of an a and a
ftelectron "spin into a "singlet state. This spinpaired state may
be expressed as a 'linear combination of the uncoupled states
in which spin 1 has orientation a and spin 2 has orientation
ft and vice versa: the singlet state is represented by
(W2)[a(1)jS(2) /3(1)a(2)].The Wigner coefficient of the
uncoupled state &(1)j3(2) is therefore 1A/2, and of the other
uncoupled state 0(1)3(2) is— 1//2. If we attempt to construct
a component of the triplet state from the two uncoupled states
we discover that taking both coefficients to be + 1// 2 would
give (1//2)[a{1)(3{2) + 0(1 )a(2)] , which should be recognized
as the M s = state of the triplet, Wigner coefficients enable all
such linear combinations to be written for the coupling of
arbitrary angular momenta into the desired resultants.
The Wigner coefficients for the coupling of a state with
quantum numbers /i ,m% and j 2 , mi into one with j,m are
written <iim l j%m 1 \jm>; a modification of this coefficient,
which being more symmetrical is easier to handle, is known as
a 3/symbol,
Further information . See Brink and Satchler (1968), Rose
(1957), and Edmonds (1957) for the properties of Wigner
coefficients. A convenient list has been published by Heine
(1960). A collection of 3/symbols in a convenient numerical
form has been prepared by Rotenberg, Bivins, Metropolis, and
Wooten (1959).
work function. The work function of a metal is the energy
required to remove an electron to infinity. The analogy with
the "ionization potential should be noticed. Metals with small
work functions can more easily lose their electrons than metals
with high work functions. A small list of work functions is
shown in Table 25.
The work function plays a role in the photoelectric effect
and in thermionic emission. The Schottky effect is the lowering
of the effective work function in the presence of an applied
electric field; this arises from the combined effect of the applied
field and the mirror charge induced by the electron as it moves
away from the surface of the metal.
Further information , A very readable account of these matters
will be found in Solymar and Walsh (1970). Comprehensive
tables of work functions are given in §9 of the American
Institute of Physics Handbook (Gray 1972). See also
"photoelectric effect.
■■■1HHBHHHBI
X
Xray spectra. Xrays are electromagnetic waves of the order
of 01 nm (1 A) wavelength. A principal terrestrial source is
the bombardment of metals with highenergy electrons. The
radiation so produced consists of two components: there is a
continuous background of radiation on which is superimposed
a sequence of sharp lines. The latter constitute the Xray
spectrum.
The continuous component, known as Bremsstrahtung. is
formed by the deceleration of the electrons by the metal: as
the negatively charged electron is decelerated when it plunges
into the metal it radiates electromagnetic radiation, and if its
initial energy is great enough there is a significant short
wavelength component.
The discrete spectral lines arise from transitions within the
corelevels of the atoms that constitute the material: the
incoming electron has enough energy to eject an innershell
electron from the atom, either completely or into some
unoccupied upper level; one of the remaining coreelectrons
falls into the hole left by the ejected electron, and the energy
difference is radiated. Highenergy ('hard') Xrays are formed
when the ejected electron comes from the Kshell (n — 1): an
electron falling from the Lshell (n  2] gives rise to thR
K a line, one falling from the Mshell (n = 3) gives the
K.line, and so on. Softer Xrays (longer wavelength) are
formed when the electron is ejected from the Lshell, and
the lines L L , etc. are formed as electrons drop from the
M and IM shells.
As a first approximation the K radiation can be treated on
the basis of the energies of the electron levels being hydrogen
like, with an effective nuclear charge of {Z — \\e, to take into
account the single electron remaining in the Isshell. Then
using the mydrogenatom energylevel formula it is an easy
matter to deduce that the frequencies of the Kradiation are
given by {Z~ 1) 2 ffl(1/1 2 )  (1/n 1 )]. Similar expressions for
other lines can be written, but they would involve different
screening constants O in Z — o. This expression shows that the
square root of the Xray frequency is proportional to the
atomic number Z: this is Moseley's law, which enabled the
elements to be put in an unambiguous order.
Questions. What different types of Xradiation can be
observed when a metal is bombarded with highenergy
electrons? What is the source of the continuous background
radiation? Why are some sharp peaks observed? To what
transitions do the peaks correspond? What is the source of
K^radiation? What is the dependence of the Xray frequency
on the atomic number of the atom? Why is it reasonable to
treat the energy calculation of Kradiation in terms of a
hydrogenlike atom with atomic number Z — 1? Calculate the
maximum frequency of the continuum Xradiation that might
be expected when a 1 keV, 100 keV, IMeV electron beam
strikes a target. The Kradiation from copper has a wavelength
of 1541 A (154 pm) and from molybdenum 0709 A(709 pm):
compute their atomic numbers. Predict the wavelength in the
case of aluminium.
Further information. For an account of Xray spectra see
Chapter XVI of White (1934), gIVC of Kuhn (1962), §iV.2 of
Herzberg (1944), and §13.9 of Condon and Shortley (1963).
Tables of X*ray transition frequencies are given in $7 of Gray
(1972).
262
z
Zee man effect. The Zeeman effect is the splitting of spectral
lines into several components by a strong magnetic field. In
the normal Zeeman effect, which is shown by atoms without
spin, each line is split into three. In the anomalous, but more
common, Zeeman effect, which is shown by atoms with net
spin, the line structure is more complicated.
In the absence of spin the only source of "magnetic
moment is the orbital angular momentum of the electrons; the
applied field interacts with the orbital moment and the energy
of the state with projection M. is changed from £ to E +
P B BM The 2L + 1 states of a term with orbital angular
momentum L are therefore no longer "degenerate but are
arrayed in a ladder with spacing p B. For example, a "P term
will be split into 3 evenly spaced components, and a l D term
into 5 components with the same splitting. The selection rule
for an optical transition is AM L = 0, ±1, and so all transitions
fall into three groups. The &M — set is at the position of
the original spectral line (see Fig. Z1), those with &M. = —1
are displaced to low frequency, and those with AM = +1 are
displaced an equal amount to high frequency. Closer analysis
of both theory and experiment shows that the light emitted is
polarized: when viewed parallel to the magneticfield direction
the &M L  line is absent, and the A/W,= ±1 lines are
circularly polarized (AfW^ = — 1 is left circularly polarized,
&M L = +t is right circularly polarized). When viewed
perpendicular to the field the AM L = line is present and
polarized parallel to the field (it is denoted a 7rline}; the
AM L — +1 lines are also planepolarized, but perpendicular to
the field (and denoted the crlines: senkrecht is German for
perpendicular). With fields of the order of 30 kG the splitting
is about T"8 cm" 1 , and this is easily detectable.
When a resultant "spin is present, so that the atom is in
some multiplet state, it is necessary to consider the effect of
the magnetic fieid on each of the "levels of the term: the
2/ + 1 states A?, of a level with total angular momentum J
have a magnetic moment (— gjH fl\)J and therefore an energy
£ + SjVqBMj in a magnetic field. The ^factor takes into
account the dependence of the magnetic moment of a state on
the magnitudes of the contributing spin and orbital angular
/
P B e
b V
+
i
\
*
'p r~\
^
V_
I
field off
field on
cr TT CT*
FIG, 21, The normal Zeeman effect ('d— *P).
263
264
Zee man effect
momenta: it is the Lande Rvalue. Since gj depends on S, L,
and J, the splitting of states is different in different terms,
and although the same selection rules apply (and the
polarizations are the same) the transitions no longer fall into
three neat groups. As an example, consider the transitions
'P — ■> 'S, The ^ term has a magnetic moment arising solely
from its spin angular momentum, and since S = 1 the field
\ 9VS
I 1
1/
t
\I_
3 > 4 tf
/
Wo
X.
h r~
\
Held off
field on
1
!
1
FIG. Z2, The anomalous Zeeman effect ( 3 P 3 S).
separates the states into three with separation 2/U B (because
9j = 2 when L = Q,S= \,J = \\. The 3 P term has three levels
%, 3 P,, and 3 Pi. Since L = 1 and S = 1, g , = for J ■ 0,
gj =  for J = 1, and f. = f for J =2. The^term is therefore
not split by the field, and the other two levels 3 Pi and 3 Pj are both
split by the field, the former into 3 states with splitting
(llMgfl and the latter into 5 states with the same splitting
(Fig. Z2). On the application of the selection rules the
spectrum is predicted to be of the form shown in Fig. Z2, and
the considerable complexity of the situation is apparent. Note
that the polarization characteristics of the lines can be used to
disentangle the spectrum. At very high fields all anomalous
Zeeman effects become normal because the field decouples the
angular momenta: this is the "PaschenBack effect.
A principal use of the Zeeman effect is the determination
of the multiplicity of terms. The splitting of energy levels by a
magnetic field is the basis of magnetic resonance techniques:
see "electron spin resonance and "nuclear magnetic resonance.
Questions. What is the Zeeman effect? Under what circum
stances are three lines seen? When does the anomalous effect
appear? Account for the normal Zeeman effect. Discuss the
Zeeman effect for the transition ' D — * ' F and construct a
diagram of the form of Fig. Z2 to illustrate the formation of
the spectrum. What is the polarization of the lines in the
'O — ► l F Zeeman spectrum? What would happen to the
polarization if the direction of the magnetic field were
reversed? What splitting would you expect in a 10 kG
magnetic field? Why does the anomalous effect depend on
the presence of spin? (Look at '^value to convince yourself
that the magnetic moment of a level of a term depends on S,
L, and J.) Construct a diagram showing the expected
anomalous Zeeman effect for the transition 3 D — ► 3 F. Mark
the polarization of the lines. What will happen to the spectrum
when the field is markedly increased (to about 100 kG>7
Further information. See MQM Chapter 8 for a more detailed
discussion. Accounts of the Zeeman effect, and the use to
which it can be put, will be found in §11.3 of Herzberg
(1944), §3.15 of King (1964), §ll I. A3 and gill. F of Kuhn
(1962), and Chapter XVI of Condon and Shortley (1963).
265
TABLE 1
Physical properties of benzene
C— C bond length
C— H bond length
enthalpy of
formation
resonance energy
first ionization
potential
polarizability
refractive index
(20°C, Dline)
relative permittivity
(20°C)
magnetic
susceptibility
AW f °
a.
1397 pm (1397 A)
1084 pm( 1084 A)
832 kJ mof 1 (19820 kcal mol" 1 )
150 kJ mof 1 (360 kcal mol" 1 )
924 eV
= 635X 10" M cm 3
(667 X 10 _4I Fm _2 )
= 1231 X 10 _24 cm 3
(1089 X 10 _4l Fm" 2 )
= 1032X 10 _24 cm 3
(913 X lO^Fm 2 )
XI.
x±
15011
= 2284
= 349X 10~ s
= 946X 10~ 5
= 747X 10~ s
absorption bands 68eV(E
lu
60eV(B lu ?<
A lg );60eV(E 2g 
A lg );49eV(B^
■v
" A lg )
TABLE 2
Bondorder— bondlength correlations
Hydrocarbon
Bond
wbond order
Bond distance (pm)
ethene
10
1335
benzene
0667
1397
naphthalene
1,2
0725
1365
2,3
0603
1404
1,9
0554
1425
9,10
0518
1393
anthracene
1,2
0738
1370
2,3
0586
1408
1,9a
0535
1423
9,9a
0606
1396
4a,9a
0485
1436
graphite
0535
1421
See Streitweiser (1961) and Daudel, Lefebvre, and Moser
(1959) for more information and analysis.
266
'5v
2C 5
TABLE 3
Character tables
C2v
(2mm)
E
c 2
o (xz)
o'M)
Ai
1
1
1
1
z
x 2 ,y 2 ,z 2
k
A 2
1
1
1
1
R :
xy
Bi
1
1
1
1
X.Ry
xz
B 2
1
1
1
1
y R x
yz
(4mm)
E
2C 4
c 2
2a
V
20 d
Ai
1
1
1
1
1
z
x 2 + y\z 2
A 2
B!
1
1
1
1
1
1
1
1
1
1
R z
x 2 y 2
B 2
1
1
1
1
1
xy
E
2
2
(x,y)(R x ,R y )
(xz, yz)
2Ci
5a
A.
A 2
Ei
E 2
1
1
2cos72°
2cos144°
1
1
2cos144°
2cos72°
1
1
z
{x,y)(R x .R y )
x 2 + y 2 ,z 2
(xz, yz)
(x 2 y 2 .xy)
2cos72
= 061803
2 cos
144
161803
(6mm)
E
2C 6
2C 3
c 2
3a
V
3 °d
A,
1
1
1
1
z
x 2 + y 2 ,z 2
A 2
1
1
1
1
R z
B,
1
—1
1
1
1
B 2
1
—1
1
1
1
E,
E 2
2
2
—1
—1
2
2
(x,yHR x ,R y )
(xz, yz)
(x 2 y 2 ,xy)
267
c, v
E
2Cf
°°a
V
A„ir
A 2 ,2T
E,.n
E 2 ,A
E 3 ,$
1
1
2
2
2
1
1
2cos0
2 cos 20
2 cos 30
1
1
z
R z
te,y)(R x ,R Y )
x 2 + y 2 ,z 2
(xz, yz)
{x 2 — y 2 ,xy)
e = exp (27TJ/3)
(x,y,z)(R x ,R v ,R z )
x 2 + y 2 + z 2
(x 2 y 2 ,2z 2 x
[xy, xz, yz)
T d
(4 3m)
E
8C 3
3C 2
6S 4
6 °d
A,
1
1
1
1
1
x 2 + y 2 + z 2
A 2
1
1
1
1
1
E
2
1
2
C2z 2 x 2 y 2 ,x 2 y 2 )
Ti
T 2
3
3
1
1
1
1
1
(xy, xz, yz)
(432)
E
8C 3
3C 2
6C 4
6C 2
Ai
1
1
1
1
1
x 2 + y 2 + z 2
A 2
1
1
1
1
1
E
2
1
2
(2z 2 x 2 y 2 ,x 2  y 2 )
T!
3
1
1
1
ix,y,z)(R x .R y ,R z )
T 2
3
1
1
1
(xy,xz,yz)
268
TABLE 4
Tvalues for selected molecules
CH 3 C0 2 H*
137
CH3I
784
CH 3 CH*0
020
CH3COCH3
791
C6H 6
273
CH3C0 2 H
793
pC 6 H(CH 3 ) 2
305
CH 3 CN
803
CgHs
431
C 2 H 2
851
C2H4
468
CeHi 2
856
H 2 (0°C)
468
C 2 H6
911
H 2
566
H 2 0(vapour)
926
C 6 H 5 0CHJ
627
CH 4
986
CHJOH
662
Si(CH 4 ) 4
1000 (definition
CgHsCHjCHs
738
HCI
1031
C6H5CH3
766
HBr
1421
CHJCHO
780
HI
2311
TABLE 5
Colour, frequency, and energy of lights
Colour
Wavelength
(nm)
Frequency
(Hz)
Wave number
(cm' 1 )
Energy
"
(eV)
(kJ mol" 1 )
(kcal mol 1 )
infrared
1000
300 X 10 14
100X 10 4
123
120
285
red
700
428
143
177
171
408
orange
620
484
161
200
193
461
yellow
580
517
172
214
206
493
green
530
566
189
234
226
539
blue
470
638
213
263
254
608
violet
420
714
238
295
285
681
near ultraviolet
300
100X 10 1S
333
415
400
957
far ultraviolet
200
150
500
622
600
143
t Adapted from Calvert and Pitts (1966).
269
TABLE 6
Dipole moments (debyes)
NH 3
147
CHCI 3
101
H
H 2
185
CH 2 CI 2
157
21
HF
191
CH3CI
187
Li
HCI
108
CH3OH
171
10
HBr
080
CH3CHO
272
Na
HI
042
CH3OCH3
130
09
HCN
30
CH 3 CH 2 OH
168
K
N0 2
032
C6H5CH3
036
o
N 2
017
oC6H4(CH 3 ) 2
062
S0 2
159
C 6 H 5 CI
157
CO
010
C 6 H 5 Br
OC6H4CI2
170
225
/77C 6 H 4 CI 2 172
TABLE 7
Pauling electronegativities
H
21
Li
Be
B
C
N
F
10
15
20
26
30
34
40
Na
Mg
Al
Si
P
S
CI
09
13
16
19
22
26
32
K
Ge
As
Se
Br
08
20
22
26
30
Rb
I
08
27
Cs
08
2
Dipole moment (in debyes) ju AB ~X A — Xq
Ionic character (per cent) 16ix A ~Xb ' + 3 ' 5l X A ~ Xb '
Covalentionic resonance energy (in eV) A~ (X A ~X e )
Mul liken scale M^ — M B = 278 (X A ~" Xg)
For a complete list of Pauling and Mulliken electronegativities
see p. 1 14 of Cotton and Wilkinson (1972).
270
TABLE 8
Oscillator strengths and molar extinction coefficients
f
e/cm 1 dm 3 mol
electric dipole allowed
1
10 4  10 5
magnetic dipole allowed
10~ 5
10" 2  10
electric quadrupole allowed
10" 7
10" 4  10" 1
spin forbidden (ST)
10" 5
10" 2  10
parity forbidden
10" 1
10 3
Examples
W nm
e/cm 1 dm 3 mol
ir**ir
C=C
180
10 000
c=cc=c
220
20 000
7T*^n
oo
280
20
N=N
350
100
c=cc=o
320
100
N=0
660
10
271
Isotope
»H
2 H
6 Li
7 Li
9 Be
"r
31 D
33o
35,
37,
41
K
79,
81
Abundance
(per cent)
999844
00156
743
9257
100
1883
8117
1108
99635
0365
0037
100
100
074
754
246
9308
691
5057
4943
TABLE 9
Hyperfine fields and spinorbit coupling in some atoms
2
Spin
2
1
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
(a.u.)
01673
2
05704
3
1408
3
2
1
2
2767
1
4770
1
2
5
2
7638
1
2
11966
56251
79187
106435
"P
(a.u.)
0775
1692
3101
4974
7546
33187
48140
67095
194127 118758 7800 456
564
This Table is adapted from Atkins and Symons (1967) and Morton, Rowlands, and Whiffen (1962).
Isotropic
coupling
<G)
508
78
39
105
130
242
725
1130
552
775
1660
17 200
3640
975
1680
1395
83
45
7800
8400
Anisotropic
coupling
(G)
38
66
34
48
104
1084
206
56
100
84
i
(cmi)
02
1
11
29
76
151
270
299
382
586
38
2460
272
TABLE 10
Diatomic molecules
Molecule
Vibration
frequency
(cm" 1 )
Bond length
(pm)
Rotation
constant
(cmi)
Dissociation
energy
(eV)
Forceconstant
(N m _1 )
Br 2 ( 79 Br 8I Br)
3232
2283
008091
1971
2458
%
164135
13117
16326
36
9521
3S CI 2
5649
1988
02438
2475
3286
12 C 16Q
217021
11281
19313
11108(?)
1902
19 F 2
8921
1435
<275
7186
'Ha
43952
7416
60809
4476
5734
2 H 2
31184
7416
30429
4553
5768
'HBr
264967
1413
8473
375
4116
1 H 3S C ,
298974
127460
105909
4430
5157
l H !9p
413852
9171
20939
<640
9655
1^27,
23095
1604
6551
3056
3141
127i
21457
2666
003735
15417
1721
14 N 2
235961
1094
2010
9756
2296
16 2
1580361
120739
144566
5080
1177
These molecules have been selected from a longer list compiled by Herzberg (1950).
273
TABLE 11
Hermite polynomials and oscillator wavefunctions
Harmonic oscillator: mass m, forceconstant k
Schrodinger equation
(h 2 /2m)d 2 0(x)/dx 2 + \kx 2 4>W = £^(x);
energies
i>= 0,1,2,...; co =(Ar//n) ,/l ;
wavefunctions
^ )= (i) M ^ ,W/21
a = mcj fh, f = / (mCob/h)x = a Vl x, W,(f) are Hermite
polynomials.
Properties of Hermite polynomials
H v ($) satisfies
Hl($)2$H' v ($) + 2vH v ($) = 0.
Rodrigues' formula
W„(f) = (1)"exp? 2 £ exp(f 2 ).
dr
Recursion relations
K® = 2VH vX®
Integrals
Cdrexp(r 2 )^(f) Wl) ,(r) = ^^^
Explicit forms
"o(f ) =
= 1
"i(?> =
= 2f
"atf) =
= 4f 2 
2
" 3 <f) =
= 8f 3 
12?
«4(J) =
= 16J*
48f 2 + 12
" 5 (?) =
= 32f 5 
160f 3 +120f
w 6 <r> =
= 64^"
480^ + 720^120
«7<J1 =
= 128f 7
1344f s + 3360f 3 
1680f
««tf) =
= 256? 8
3584$* + 13440? 4 
13440f 2 + 1680
TABLE 12
Some Debye temperatures of solids
d D /K
d D /K
e D /K
Li
344
Cu
343
Al
428
Na
156
Ag
226
Ti
420
K
911
Au
162
Hg
719
Rb
555
Cs
395
C (diamond)
2230
NaCI
321
KCI
230
KBr
177
CaF 2
570
274
Coordination
number
2
3
TABLE 13
Hybrid orbitals
Shape
linear
bent
trigonal planar
unsymmetrical planar
trigonal pyramidal
tetrahedral
irregular tetrahedral
tetragonal pyramidal
bi pyramidal
tetragonal pyramidal
pentagonal planar
pentagonal pyramidal
octahedral
trigonal prismatic
trigonal antiprismatic
Hybridization
sp, dp
p 2 , ds, d 2
sp 2 , dp 2 , ds 2 , d 3
dsp
P 3 .d 2 P
sp 3 , d 3 s
d 2 sp, dp 3 , d 3 p
d 4
dsp 3 , d 3 sp
d 2 sp 2 , d 4 s, d 2 p 3 , d 4 p
d 3 p 2
d s
d 2 sp 3
d 4 sp, d 5 p
d 3 p 3
TABLE 14
Associated Laguerre polynomials and functions
The Laguerre differential equation is
x(d 2 f/dx 2 ) + (1 x){df/dx) +nf= 0,
with n a nonnegative integer. Its solutions are the Laguerre
polynomials L n (x),
^=M,»!(tfi')(^)v
= e*(d/dx)"x'V x .
The associated Laguerre polynomials are related to L n (x) by
differentiation:
and satisfy
x(d 2 f/dx 2 ) + (k + 1 x)(df/dx) + in  k)f =
The associated Laguerre functions are related to L k (x) as
L k U) = d k L n (x)/dx k
J*{x) = e y ' x x ,Mk  1) L k „{x)
follows:
 n r~,  „.
and satisfy the equation
xldV/dx 2 ) + 2(df/dx) + [n±(k 1)  \x  (k 2  1)/4x]^=0.
The normalized radial components of the hydrogenatom wave
functions are
with p = 2Zr/na Q
These wavef unctions are developed in Table 15.
(P),
275
TABLE 15
Hydrogen atom wavefunctions
General form:
R n(r) are proportional to the associated Laguerre functions
(see Table 14) and the Yn (0, <j>) are the spherical harmonics
(see Table 23).
Specific form of radial equation; p = 2Zr/na .
1s: R 10 (r) = IZ/aoJ^af" 3
2s: R 20 {r) = (Z/a ) 3/2 (1/2/2)(2p)e p/2
2p: /? 21 (r) = (Z/a ) 3/2 (1/2/6)pe" /2
3s: /?3o(/) = (Z/a ) 3/2 (1/9/3)(6 6p + p 2 )e" p/2
3p: /?3i(r) = (Z/a ) 3/2 (1/9/61(4 pipe"" 72
3d: R 32 (r) = (Z/a ) 3/2 (1/9/30)pV p/2
4s: /? 40 (r) = (Z/a ) 3/2 (1/96)(24 36p+ 12p 2 p 3 )e"" p/2
4p: /? 41 (r) = (Z/a ) V2 ( 1/32/ 15) (201 Op + p 2 )pe _p/2
4d: /? 42 (r) = (Z/a ) 3/2 ( 1/96/ 5) (6 p)pV> /2
4f: R 43 (r) = (Z/a ) 3/2 (1/96/35) pV p/2
276
TABLE 16
The hydrogen atom
Experimental data
Spectral lines (Vnm):
Lyman series: 121567(a), 102572(/3), 97253(7) 9115
Balmer series: 65628(a), 48613(j8), 43405(7) 3646
Paschen series: 18751, 12818, 10938, . . ., 8204
Brackett series: 40512, 26251 14584
Pfund series: 74512 . . ., 22788
Humphreys series: 12 368, . . ., 32814.
Ionization potential: 1097 X 10 s cm 1 , 1360 eV.
Electron affinity: 077 eV.
Lamb shift: (2S V 2P„): 1058 GHz.
At Vi
Hyperfine interaction (Fermi contact interaction):
14204 MHz.
Polarizability (ground state): 45 a%, 87 X 10~ 31 m 3 .
Diamagnetic susceptibility (ground state): —397 X 10" 6 .
Covalent radius: 30 pm (030 A).
Electronegativity (Pauling): 21.
Theoretical data
Hamiltonian for the atom: H = — (h 2 /2p)V 2 — e 2 /47re r.
Energy of state with quantum numbers n, 2, m»:
E o _ / Pe< \ 1 _ An
n= 1,2, 3, ...;£ = 0, 1,2 /71;/t? 6 = £, £1 £.
p is the reduced mass m m Hm + m ) and /?„ is the
•^ e p e p H
"Rydberg constant.
Degeneracy of state with energy — R In 2 : n 2 .
Wavefunctions ty nim Ar'. 9. </>) = R n %(r) Y^ (0, 0).
The angular functions are the "spherical harmonics, see
Table 23; the radial functions are the associated Laguerre
functions
R ni 1r) «e'V A 2 « +1
(P)
where p = 2Zr/na and L (p) is an associated Laguerre
polynomial. See Tables 14 and 15 for their analytical form
and normalization.
Expectation values, etc.
Mean radius, etc: t
(r 2 )=(aW/Z 2 )h +
3 (, 3£(£+1)A
l(1^i)j
<r)={aon 2 /Z) J1+,
<r 1 ) = Z/a n 2
<r 2 > = Z 2 /agA7 3 (£ + l)
<r 3 >=(Z/a ) 3 /n 3 £(£ + )(£+1).
Most probable radius (ground state): a (Bohr radius,
5291 771 5X 10" n m).
Spinorbit coupling parameter:
\87re /7? 2 c 2 ao
u 2 z*Rjr 2
/7 3 £(£+j)(£+1) '
a is the "finestructure constant.
Probability at the nucleus: \\jj (0)l 2 :
/? 3 £(£ + 5)(£+ 1).
■Z 3 /ira 3 n 3 .
277
Isotope
(*: radioactive)
l H
3 H*
y Be
io D
14,
N
N
is,
19 F
23,
Na*
Na
3S S*
35 CI
36 C .
37 CI
39 K
40 K*
43,
79,
'Ca
Br
*Br
TABLE 17
Selected nuclearspin properties
Natural
abundance
(per cent)
Spin
Magnetic
moment

1
2
19130
999844
1
2
279270
156 X 10" 2
1
085738

1
2
29788
743
1
082191
9257
3
2
32560
100
3
2
11774
1883
3
18006
8117
3
2
26880
1108
1
2
070216
99635
1
040357
0365
1
2
028304
37 X 10~ 2
5
2
18930
100
1
2
26273

3
1745
100
3
2
22161
100
1
2
11305
074
3
2
064274

3
2
100
754
3
2
082089

2
12838
246
3
2
068329
9308
3
2
039094
119 X 10~ 2
4
1294
691
3
2
021453
013
7
2
13153
5057
3
2
20990
4943
3
2
22626
Electric
quadrupole
moment
(e X 1024cm~2)
277 X 10~ 3
46 X 10" 4
42 X 10~ 2
2X 10" 2
0111
355 X 10~ 2
2X 10~ 2
4X 10" 3
01
64 X 10~ 2
45 X 10~ 2
797 X 10" 2
168X 10" 2
621 X 10" 2
?
?
?
?
033
028 .
n.m.r. frequency
at 10 kG
(MHz)
29165
42577
6536
45414
6265
16547
5983
4575
13660
10705
3076
4315
5772
40055
4434
11262
17235
3266
508
4172
4893
3472
1987
2470
1092
2865
10667
11498
278
TABLE 18
First and second ionization potentials (in eV) of some elements
H
13599
He
24588
54418
Li
5392
75641
Be
9323
18211
B
8298
25156
C
11260
24383
N
1453
29602
13618
35118
F
17423
3498
Ne
21565
40964
Na
5139
4729
Mg
7646
15035
Al
5986
18828
Si
8152
16346
P
10487
1972
S
10360
234
CI
12967
2380
Ar
15760
2762
K
4341
3181
Ca
6113
11872
Br
11814
216
Kr
14000
2456
279
Gas lasers
HeNe
Ne
Ar
Kr
Xe
H 2
N 2
C0 2
Solidstate lasers
Ruby
Nd 3+ :YAG
GaAs
In As
TABLE 19
Some laser systems
\/nm
Power
Mode and duration of pulse
6328
100 mW
CW
1084
10 mW
CW
1152
20 mW
CW
3391
10 mW
CW
5401
10 kW
pulsed (3 ns)
4880
10W
CW
5145
500 mW
CW
4579
40 mW
CW
4658

CW
4727
—
CW
4765
100 mW
CW
4965
100 mW
CW
6471
5W
CW
5682
50 mW
CW
5208
50 mW
CW
4762
25 mW
CW
1700
—
1160
—
3371
02 MW
pulsed (10 ns)
10600 (106 Mm)
1 kW
CW
6943
400 kW
conventional pulsed (1 ms
16 GW
Qswitched (1020 ns)
16 GW
modelocked (10 ps)
10600
300 W
CW
10 MW
Qswitched (10 ns)
fOMW
modelocked (1 ps)
9050
60 W
pulsed (200 ns)
8450
1 W
pulsed (2 jus)
3150
50 mW
pulsed (2 ns)
280
TABLE 20
Maxwell equations
Basic definitions
E : electricfield intensity (V m _1 )
H : magneticfield intensity (Am 1 )
D : electric displacement (C m~ 2 )
B : magnetic induction, flux density (T or Wb rrf 2 )
p : charge density (C m" 3 )
J : current density (A m 2 ).
D and B may be related to E and H respectively, through the
polarization P and magnetization M:
D = e E + P B = jUoH + /UoM (/ioe = c 2 ).
Maxwell equations
V.D =p
V.B =0
VAE = bB/bt
VAH = J + dD/dt.
Potentials
A : vector potential, : scalar potential
B = VAA
E = dA/df  V0.
Gauge transformation : if A — ■> A + Vx and — *■ — dx/3 i.
where x is any differentiable function, E and B are unchanged.
When x is chosen so that V.A = we are in the Coulomb gauge
and if V.A + (1/c 2 )(90/3r) = we are in the Lorentz gauge.
TABLE 21
Slater atomic orbitals
Values of Z_, x
eff
= Z—O for s.
porbitals of the neutral first
and secondrow atoms.
1s
H
1
He
170
Li
Be
B
C
N
F
Ne
1s
270
370
470
570
670
770
870
970
2s,2p
130
195
260
325
390
455
520
585
Na
Mg
Al
Si
P
S
CI
Ar
Is
1070
1170
1270
1370
1470
1570
1670
1770
2s, 2p
685
785
885
985
1085
1185
1285
1385
3s, 3p
220
285
350
415
480
545
610
675
281
TABLE 22
Spherical harmonics and Legendre functions
The spherical harmonics VV<0, 0) satisfy
A 2 ^ =«<£ + D/^ ^VA'.'.'.e.
Factorization
/^ are the associated Legendre functions:
^(x) = [(1x 2 ) m/2 /2 £ C!
<x 2 1)*
(1 x 2 )"
dx" 1
*>(*>
1) c .
Pj are the Legendre functions:
/' B (x) = (1/2 B £l)f i iJUf 2 
Properties of Vn
symmetry:
y^ (7T 0, + ir) = (1 ) £ v^ (0, 0)
orthonormality:
;; d 0sin0 £* d^ v. w^id, <t» = o^s^
(5no'5 ' = 1 if both 2 = £' and m = m ', and otherwise),
xx mm
Recursion relations for Pg":
(£/n + D/^ +1 <x)  (2B + Dx/^W + (« + mV»g! I U) =
x/f(x)  (£ /n + Dd x 2 )*Af _1 (x) /»£, (x) =
/f tl M xP^U)  (C + m)d x 2 ) 14 ^" 1 (x) =
(£/n + D/^ +1 W + (1 x 2 )*Ff +l U) 
{C + /r7 + 1)x/' c T '(x) =
(1 x 2 ) 34 /^* 1 (x)  Zrox/'g 1 (x) +
+ (x + m)(xm+1)(1x 2 ) , V(r l (x) =
(1 x 2 )* m/fM = (C + Dx/fU)  (fim + DPg^lx)
= (fi + m)Pg: i (x)ex/ , K "{x).
Integrals of P™\
f^dxfffWPftb) = 26 M »[(£ + m)\IW + 1)(« /n)l]
282
£
1
m
±1
±1
±2
±1
±2
±3
TABLE 23
Cartesian and polar forms of spherical harmonics VV,
Cartesian form
1/27I 54
jO/n)* <*//■)
4(3/27T) ,/ '(x±iK)/r
i(5/7r)' /2 (2z 2 x 2 /V 2
+~mi2n) y *z(x±\y)lr 2
£(15/27r) ya (x±iy) 2 /r 2
5(7/7r) ya z(2? 2 3x 2 3y 2 )// 3
+(21/7r) ,/a (x ± \y)(4z 2 x 2 y 2 )/r 3
£(105/27T) ,/a z(x iiy) 2 // 3
4(35/7r) ,/a (x+i/) 3 // 3
Polar form
1/27I 34
5<3/7T) y2 cos0
1
_(3/27r)' /a sin0e ±i0
j(5/7r) ,/j (3cos 2 01)
+j(15/27r) ,/a cos0sin0e ±i0
£<15/27r) ya sin 2 0e ±2i *
j(7/7r)' /a (2cos 3  3cos0 sin 2 0)
4(21/7T)' /a (4cos 2 sin0  sin 3 0)e ±i0
£(1 05/27^0)50 sin 2 0e ±2i0
4(35/ir) ,/a sin 3 0e ±3i *
283
TABLE 24
Vibration frequencies
CH stretch
28502960 cm" 1
CHbend
13401465
C— C bend and stretch
7001250
C=C stretch
16201680
C^C stretch
21002260
0— H stretch
35903650
H— bonds
32003570
C=0 stretch
16401780
N— H stretch
32003500
C^N stretch
22152275
C=N stretch
14801690
N=N stretch
15751630
P— H stretch
23502440
CF stretch
10001400
CCI stretch
600800
C— Br stretch
500600
C— 1 stretch
~500
cor
14101450
N0~ 3
13501420
NO2
12301250
sor
10801130
silicates
9001100
See Bellamy (1958, 1968) for <
i thorough analysis.
TABLE 25
Photoelectric (and thermionic) work functions (eV)
Li
Be
242
39
(367)
Na
Mg
23
37
K
Cu
Ca
Zn
225
48
32
43
(45)
(22)
Rb
Ag
Cd
209
43
(43)
41
Cs
Au
214
54
(186)
(43)
284
TABLE 26
TABLE 27
Covalent radii
Vector properties
Element
fl/pm
We deal with the vectors F and G, where
H
c
C=
30
77
67
F = F \ + F \ + F k
x y' z
G = G x i + G / j + G z k.
C=
60
Scalar product
C— (benzene)
o
0=
695
66
57
F.G = FG +F G +F G .
xx y y z z
Vector product
N— (amino)
N— (nitrate)
N= (nitrate)
70
70 •
65
FAG =
i j k
F F F
x y z
G G G
x y z
= GA F.
s
S= (sulphate)
ci—
104
95
99
Triple and quadruple products
F.(G A H) = G.(H A F) = H.(F A G) = (F A G).H
Br
114
F A (G A H) = G(F.H)  H(F.G)
1
133
(F A G) . (H A 1) = (F.H)(G.I)  (F.I)(G.H) .
divergence
divF = V.F=(^j + ^ + ^, a scalar.
curl
1 i J k 1
curl F = V A F = 1 3/3* 3/3/ 3/3*  , a vector.
\ F x F y
F z \
differentiation of products:
V.(UF) = UV.F + F.Vt/
V A (i/F) = L/(V A F) + (VLO A F
V.(F A G) = G.(V A F)  F.(V A G)
\7A(VAF) =V(V.F)\7 2 F
V A (F A G) = F(V.G)  (V.F)G + (G.V)F  (F.V)G
V(F.G) = (F.V)G + (G.V)F + F A (V A G) + G A (V A F)
V 2 U= V.VU
V A (Vi/) = .
Bibliography
MQM refers to
ATKINS, P.W. (1970). Molecular quantum mechanics. Clarendon Press,
Oxford.
Volumes in the Oxford Chemistry Series (denoted OCS) which are now
available and which elaborate many of the points in the text, are as
follows:
1. McLAUCHLAN, K.A. (1972). Magnetic resonance.
2. ROBBINS, J. (1972). Ions in solution (2): an introduction to
electrochemistry.
3. PUDDEPHATT, R.J. (1972). The periodic table of the elements.
4. JACKSON, R.A. (1972). Mechanism: an introduction to the study
of organic reactions.
5. WHITTAKER, D. (1973). Stereochemistry and mechanism.
6. HUGHES, G. (1973). Radiation chemistry.
7. PASS, G. (1973). Ions in solution (3): inorganic properties.
8. SMITH, E.B. (1973). Basic chemical thermodynamics.
9. COULSON, C.A. (1973). 77>e shape and structure of molecules.
10. WORMALD, J. (1973). Diffraction methods.
11. SHORTER, J. (1973). Correlation analysis in organic chemistry:
an introduction to linear freeenergy relationships.
12. STERN, E.S. (Ed.) (1973). The chemist in industry (1): fine
chemicals for polymers.
13. EARNSHAW, A. and HARRINGTON, T.J. (1973). The chemistry
of the transition elements.
14. ALBERY, W.J. (1974). Electrode kinetics.
16. FYFE, W.S. (1974). Geochemistry.
17. STERN, E.S. (Ed.) (1974). The chemist in industry (2): human
health and plant protection.
18. BONO, G.C. (1974). Heterogeneous catalysis: principles and
applications.
19. GASSER, R.P.H. and RICHARDS, W.G. (1974). Entropy and
energy levels.
20. SPEDDING, D.J. (1974). Air pollution.
21. ATKINS, P.W. (1974). Quanta: a handbook of concepts.
22. PILLING, M.J. (1974). Reaction kinetics.
Volumes in the Oxford Physics Series (OPS) now available are
1. ROBINSON, F.N.H. (1973). Electromagnetism.
2. LANCASTER, G. (1973). D.c. and a.c. circuits.
3. INGRAM, D.J.E. (1973). Radiation and quantum physics.
5. JENNINGS, B.R. and MORRIS, V.J. (1974). Atoms in contact
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